mixedcos

Percentage Accurate: 66.9% → 98.2%

Time: 10.6s

Alternatives: 16

Speedup: 9.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(c\_m \cdot s\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= c_m 9.5e-230)
     (/ (cos (* 2.0 x)) (pow (* x (* c_m s_m)) 2.0))
     (/ (/ (cos (+ x x)) t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (c_m <= 9.5e-230) {
		tmp = cos((2.0 * x)) / pow((x * (c_m * s_m)), 2.0);
	} else {
		tmp = (cos((x + x)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if (c_m <= 9.5d-230) then
        tmp = cos((2.0d0 * x)) / ((x * (c_m * s_m)) ** 2.0d0)
    else
        tmp = (cos((x + x)) / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (c_m <= 9.5e-230) {
		tmp = Math.cos((2.0 * x)) / Math.pow((x * (c_m * s_m)), 2.0);
	} else {
		tmp = (Math.cos((x + x)) / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if c_m <= 9.5e-230:
		tmp = math.cos((2.0 * x)) / math.pow((x * (c_m * s_m)), 2.0)
	else:
		tmp = (math.cos((x + x)) / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (c_m <= 9.5e-230)
		tmp = Float64(cos(Float64(2.0 * x)) / (Float64(x * Float64(c_m * s_m)) ^ 2.0));
	else
		tmp = Float64(Float64(cos(Float64(x + x)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if (c_m <= 9.5e-230)
		tmp = cos((2.0 * x)) / ((x * (c_m * s_m)) ^ 2.0);
	else
		tmp = (cos((x + x)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 9.5e-230], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[Power[N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 9.5000000000000004e-230

   1. Initial program 62.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   if 9.5000000000000004e-230 < c

   1. Initial program 59.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.6

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 97.6

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 96.5

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{x \cdot \left(c\_m \cdot \left(s\_m \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0)))))
        -1e-187)
     (/ (fma x (* x -2.0) 1.0) (* x (* c_m (* s_m t_0))))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -1e-187) {
		tmp = fma(x, (x * -2.0), 1.0) / (x * (c_m * (s_m * t_0)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -1e-187)
		tmp = Float64(fma(x, Float64(x * -2.0), 1.0) / Float64(x * Float64(c_m * Float64(s_m * t_0))));
	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]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-187], N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * N[(c$95$m * N[(s$95$m * t$95$0), $MachinePrecision]), $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))) < -1e-187

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 23.8

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

   7. Applied rewrites 23.8%

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

   if -1e-187 < (/.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 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. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   7. Applied rewrites 86.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{x \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{t\_1}}{t\_1}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+229}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(c\_m \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* c_m (* x s_m))))
   (if (<= x 1350000000.0)
     (/ (/ (fma x (* x -2.0) 1.0) t_1) t_1)
     (if (<= x 5.8e+229)
       (/ t_0 (* s_m (* c_m (* x (* x (* c_m s_m))))))
       (/ t_0 (* x (* c_m (* s_m (* s_m (* c_m x))))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 1350000000.0) {
		tmp = (fma(x, (x * -2.0), 1.0) / t_1) / t_1;
	} else if (x <= 5.8e+229) {
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	} else {
		tmp = t_0 / (x * (c_m * (s_m * (s_m * (c_m * x)))));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 1350000000.0)
		tmp = Float64(Float64(fma(x, Float64(x * -2.0), 1.0) / t_1) / t_1);
	elseif (x <= 5.8e+229)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(x * Float64(x * Float64(c_m * s_m))))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(c_m * Float64(s_m * Float64(s_m * Float64(c_m * x))))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1350000000.0], N[(N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 5.8e+229], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e9

   1. Initial program 60.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 96.7

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 96.7

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 95.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.7

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

   9. Applied rewrites 74.7%

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

   if 1.35e9 < x < 5.79999999999999963e229

   1. Initial program 71.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.5

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 97.5

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 97.6

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. count-2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 95.9

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

      17. lift-*.f64 N/A

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

      18. count-2 N/A

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

      19. lift-+.f64 95.9

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

      20. lift-pow.f64 N/A

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

   8. Applied rewrites 89.8%

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

   if 5.79999999999999963e229 < x

   1. Initial program 48.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

      1. count-2 N/A

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

      2. lift-+.f64 90.8

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

   8. Applied rewrites 90.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+229}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{t\_1}}{t\_1}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x \cdot \left(c\_m \cdot \left(s\_m \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* c_m (* x s_m))))
   (if (<= x 1350000000.0)
     (/ (/ (fma x (* x -2.0) 1.0) t_1) t_1)
     (if (<= x 2e+230)
       (/ t_0 (* s_m (* c_m (* x (* x (* c_m s_m))))))
       (/ t_0 (* x (* c_m (* s_m t_1))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 1350000000.0) {
		tmp = (fma(x, (x * -2.0), 1.0) / t_1) / t_1;
	} else if (x <= 2e+230) {
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	} else {
		tmp = t_0 / (x * (c_m * (s_m * t_1)));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 1350000000.0)
		tmp = Float64(Float64(fma(x, Float64(x * -2.0), 1.0) / t_1) / t_1);
	elseif (x <= 2e+230)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(x * Float64(x * Float64(c_m * s_m))))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(c_m * Float64(s_m * t_1))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1350000000.0], N[(N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 2e+230], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(c$95$m * N[(s$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e9

   1. Initial program 60.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 96.7

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 96.7

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 95.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.7

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

   9. Applied rewrites 74.7%

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

   if 1.35e9 < x < 2.0000000000000002e230

   1. Initial program 71.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.5

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 97.5

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 97.6

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. count-2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 95.9

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

      17. lift-*.f64 N/A

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

      18. count-2 N/A

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

      19. lift-+.f64 95.9

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

      20. lift-pow.f64 N/A

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

   8. Applied rewrites 89.8%

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

   if 2.0000000000000002e230 < x

   1. Initial program 48.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

      1. count-2 N/A

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

      2. lift-+.f64 86.2

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

   6. Applied rewrites 86.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s\_m \cdot \left(\left(c\_m \cdot x\right) \cdot \left(s\_m \cdot \left(c\_m \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 3.2e+14)
     (/ (/ (cos (+ x x)) t_0) t_0)
     (/ (cos (* 2.0 x)) (* s_m (* (* c_m x) (* s_m (* c_m x))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 3.2e+14) {
		tmp = (cos((x + x)) / t_0) / t_0;
	} else {
		tmp = cos((2.0 * x)) / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if (x <= 3.2d+14) then
        tmp = (cos((x + x)) / t_0) / t_0
    else
        tmp = cos((2.0d0 * x)) / (s_m * ((c_m * x) * (s_m * (c_m * x))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 3.2e+14) {
		tmp = (Math.cos((x + x)) / t_0) / t_0;
	} else {
		tmp = Math.cos((2.0 * x)) / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if x <= 3.2e+14:
		tmp = (math.cos((x + x)) / t_0) / t_0
	else:
		tmp = math.cos((2.0 * x)) / (s_m * ((c_m * x) * (s_m * (c_m * x))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 3.2e+14)
		tmp = Float64(Float64(cos(Float64(x + x)) / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(s_m * Float64(Float64(c_m * x) * Float64(s_m * Float64(c_m * x)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if (x <= 3.2e+14)
		tmp = (cos((x + x)) / t_0) / t_0;
	else
		tmp = cos((2.0 * x)) / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.2e+14], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(c$95$m * x), $MachinePrecision] * N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e14

   1. Initial program 60.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 96.8

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 96.8

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 95.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if 3.2e14 < x

   1. Initial program 64.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 86.4

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

   6. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 91.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 92.5

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

   10. Applied rewrites 92.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 22500000000:\\ \;\;\;\;\frac{t\_0}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(\left(c\_m \cdot x\right) \cdot \left(s\_m \cdot \left(c\_m \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (* c_m (* x s_m))))
   (if (<= x 22500000000.0)
     (/ t_0 (* t_1 t_1))
     (/ t_0 (* s_m (* (* c_m x) (* s_m (* c_m x))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 22500000000.0) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = c_m * (x * s_m)
    if (x <= 22500000000.0d0) then
        tmp = t_0 / (t_1 * t_1)
    else
        tmp = t_0 / (s_m * ((c_m * x) * (s_m * (c_m * x))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 22500000000.0) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	t_1 = c_m * (x * s_m)
	tmp = 0
	if x <= 22500000000.0:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = t_0 / (s_m * ((c_m * x) * (s_m * (c_m * x))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 22500000000.0)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(Float64(c_m * x) * Float64(s_m * Float64(c_m * x)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((2.0 * x));
	t_1 = c_m * (x * s_m);
	tmp = 0.0;
	if (x <= 22500000000.0)
		tmp = t_0 / (t_1 * t_1);
	else
		tmp = t_0 / (s_m * ((c_m * x) * (s_m * (c_m * x))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 22500000000.0], N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(N[(c$95$m * x), $MachinePrecision] * N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e10

   1. Initial program 60.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.7

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 95.3

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 2.25e10 < x

   1. Initial program 63.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 87.0

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

   4. Applied rewrites 87.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 85.6

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

   6. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 90.2%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 91.5

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

   10. Applied rewrites 91.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 22500000000:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 8.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(\left(c\_m \cdot x\right) \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= c_m 8.2e-230)
     (/ (cos (+ x x)) (* s_m (* (* c_m x) (* x (* c_m s_m)))))
     (/ (cos (* 2.0 x)) (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (c_m <= 8.2e-230) {
		tmp = cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))));
	} else {
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if (c_m <= 8.2d-230) then
        tmp = cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))))
    else
        tmp = cos((2.0d0 * x)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (c_m <= 8.2e-230) {
		tmp = Math.cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))));
	} else {
		tmp = Math.cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if c_m <= 8.2e-230:
		tmp = math.cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))))
	else:
		tmp = math.cos((2.0 * x)) / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (c_m <= 8.2e-230)
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(Float64(c_m * x) * Float64(x * Float64(c_m * s_m)))));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if (c_m <= 8.2e-230)
		tmp = cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))));
	else
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 8.2e-230], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(c$95$m * x), $MachinePrecision] * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 8.2000000000000003e-230

   1. Initial program 62.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 88.4

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

   6. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 87.5%

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

      1. count-2 N/A

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

      2. lift-+.f64 87.5

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

   10. Applied rewrites 87.5%

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

   if 8.2000000000000003e-230 < c

   1. Initial program 59.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.8

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 95.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(\left(c \cdot x\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(\left(c\_m \cdot x\right) \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 1350000000.0)
     (/ (/ (fma x (* x -2.0) 1.0) t_0) t_0)
     (/ (cos (+ x x)) (* s_m (* (* c_m x) (* x (* c_m s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 1350000000.0) {
		tmp = (fma(x, (x * -2.0), 1.0) / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (s_m * ((c_m * x) * (x * (c_m * s_m))));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 1350000000.0)
		tmp = Float64(Float64(fma(x, Float64(x * -2.0), 1.0) / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(Float64(c_m * x) * Float64(x * Float64(c_m * s_m)))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1350000000.0], N[(N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(c$95$m * x), $MachinePrecision] * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e9

   1. Initial program 60.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 96.7

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 96.7

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 95.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.7

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

   9. Applied rewrites 74.7%

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

   if 1.35e9 < x

   1. Initial program 63.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 87.0

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

   4. Applied rewrites 87.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 85.6

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

   6. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 90.2%

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

      1. count-2 N/A

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

      2. lift-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(\left(c \cdot x\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 9.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(\left(c\_m \cdot s\_m\right) \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 9.8e-34)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x x)) (* x (* (* c_m s_m) (* x (* c_m s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 9.8e-34) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if (x <= 9.8d-34) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x + x)) / (x * ((c_m * s_m) * (x * (c_m * s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 9.8e-34) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x + x)) / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if x <= 9.8e-34:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x + x)) / (x * ((c_m * s_m) * (x * (c_m * s_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 9.8e-34)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(x * Float64(Float64(c_m * s_m) * Float64(x * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if (x <= 9.8e-34)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x + x)) / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.8e-34], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.79999999999999925e-34

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   7. Applied rewrites 85.7%

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

   if 9.79999999999999925e-34 < x

   1. Initial program 64.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x}} \]
   5. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]

      2. lift-+.f64 87.9

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]

   6. Applied rewrites 87.9%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s\right)\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot s\right)\right) \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)}\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right) \cdot c\right)} \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)} \cdot c\right) \cdot x} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot c\right)\right)} \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot s\right)}\right) \cdot x} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot s\right)}\right) \cdot x} \]

      9. lower-*.f64 89.1

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot s\right)\right)} \cdot x} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot s\right)\right) \cdot x} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right) \cdot x} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      14. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(c \cdot s\right)\right) \cdot x} \]

      15. lift-*.f64 91.9

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(c \cdot s\right)\right) \cdot x} \]

   8. Applied rewrites 91.9%

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 1350000000.0)
     (/ (/ (fma x (* x -2.0) 1.0) t_0) t_0)
     (/ (cos (+ x x)) (* s_m (* c_m (* x (* x (* c_m s_m)))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 1350000000.0) {
		tmp = (fma(x, (x * -2.0), 1.0) / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 1350000000.0)
		tmp = Float64(Float64(fma(x, Float64(x * -2.0), 1.0) / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(c_m * Float64(x * Float64(x * Float64(c_m * s_m))))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1350000000.0], N[(N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(c$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e9

   1. Initial program 60.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 96.7

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 96.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}} \]

      8. lower-/.f64 96.7

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}}{\left(c \cdot s\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      10. count-2 N/A

          \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      11. lift-+.f64 96.7

          \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot x}}}{\left(c \cdot s\right) \cdot x} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right)} \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      16. lift-*.f64 95.3

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{\left(c \cdot s\right)} \cdot x} \]

      19. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \]

      21. lift-*.f64 98.0

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

   6. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      7. lower-*.f64 74.7

         \[\leadsto \frac{\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

   9. Applied rewrites 74.7%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

   if 1.35e9 < x

   1. Initial program 63.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 97.2

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 97.2%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}} \]

      8. lower-/.f64 98.3

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}}{\left(c \cdot s\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      10. count-2 N/A

          \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      11. lift-+.f64 98.3

          \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot x}}}{\left(c \cdot s\right) \cdot x} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right)} \cdot x}}{\left(c \cdot s\right) \cdot x} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      16. lift-*.f64 95.3

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot s\right) \cdot x} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{\left(c \cdot s\right)} \cdot x} \]

      19. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \]

      21. lift-*.f64 96.7

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

   6. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]
   7. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\cos \left(2 \cdot x\right)}}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \color{blue}{\left(s \cdot x\right)}}}{c \cdot \left(s \cdot x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}}}{c \cdot \left(s \cdot x\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot x\right)}}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

      8. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

      9. pow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}}^{2}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      16. lower-/.f64 97.2

          \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      18. count-2 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      19. lift-+.f64 97.2

          \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      20. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1350000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c\_m \cdot \left(s\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 6.2e-43)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x x)) (* c_m (* s_m (* x (* x (* c_m s_m)))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 6.2e-43) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (c_m * (s_m * (x * (x * (c_m * s_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if (x <= 6.2d-43) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x + x)) / (c_m * (s_m * (x * (x * (c_m * s_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 6.2e-43) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x + x)) / (c_m * (s_m * (x * (x * (c_m * s_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if x <= 6.2e-43:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x + x)) / (c_m * (s_m * (x * (x * (c_m * s_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 6.2e-43)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(c_m * Float64(s_m * Float64(x * Float64(x * Float64(c_m * s_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if (x <= 6.2e-43)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x + x)) / (c_m * (s_m * (x * (x * (c_m * s_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.2e-43], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(c$95$m * N[(s$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.1999999999999999e-43

   1. Initial program 60.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.0

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      13. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   7. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

   if 6.1999999999999999e-43 < x

   1. Initial program 63.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot {s}^{2}\right)\right)\right)} \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot {s}^{2}\right)\right)\right)} \cdot x} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(c \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right)\right) \cdot x} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(c \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)\right)\right) \cdot x} \]

      12. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(c \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \cdot x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}\right)\right) \cdot x} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \color{blue}{\left(\left(c \cdot \left(x \cdot s\right)\right) \cdot s\right)}\right) \cdot x} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \color{blue}{\left(\left(c \cdot \left(x \cdot s\right)\right) \cdot s\right)}\right) \cdot x} \]

      16. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s\right)\right) \cdot x} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot s\right)\right) \cdot x} \]

      18. lower-*.f64 87.2

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s\right)\right) \cdot x} \]

   4. Applied rewrites 87.2%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x}} \]
   5. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]

      2. lift-+.f64 87.2

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]

   6. Applied rewrites 87.2%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)\right) \cdot x} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right) \cdot c\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)} \cdot c\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(s \cdot \left(\left(s \cdot {x}^{2}\right) \cdot c\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(s \cdot {x}^{2}\right)\right)\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot x\right)}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot x\right)}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \color{blue}{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \color{blue}{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      18. lower-*.f64 87.2

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   9. Applied rewrites 87.2%

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(s \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(x \cdot \left(c\_m \cdot \left(s\_m \cdot \left(x \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<= x 5.5e-163)
   (/ 1.0 (* c_m (* x (* c_m (* s_m (* x s_m))))))
   (/ 1.0 (* c_m (* c_m (* s_m (* s_m (* x x))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if (x <= 5.5e-163) {
		tmp = 1.0 / (c_m * (x * (c_m * (s_m * (x * s_m)))));
	} else {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 5.5d-163) then
        tmp = 1.0d0 / (c_m * (x * (c_m * (s_m * (x * s_m)))))
    else
        tmp = 1.0d0 / (c_m * (c_m * (s_m * (s_m * (x * x)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if (x <= 5.5e-163) {
		tmp = 1.0 / (c_m * (x * (c_m * (s_m * (x * s_m)))));
	} else {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if x <= 5.5e-163:
		tmp = 1.0 / (c_m * (x * (c_m * (s_m * (x * s_m)))))
	else:
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (x <= 5.5e-163)
		tmp = Float64(1.0 / Float64(c_m * Float64(x * Float64(c_m * Float64(s_m * Float64(x * s_m))))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(s_m * Float64(x * x))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if (x <= 5.5e-163)
		tmp = 1.0 / (c_m * (x * (c_m * (s_m * (x * s_m)))));
	else
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[x, 5.5e-163], N[(1.0 / N[(c$95$m * N[(x * N[(c$95$m * N[(s$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999998e-163

   1. Initial program 59.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 69.7

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right) \cdot x}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x} \]

      10. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(s \cdot x\right) \cdot \left(\left(c \cdot \left(c \cdot s\right)\right) \cdot x\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot s\right)\right)} \cdot x\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      14. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      18. lower-*.f64 78.5

          \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      21. associate-*r* N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

   7. Applied rewrites 80.9%

      \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right) \cdot x\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)} \cdot x\right)} \]

      8. lower-*.f64 77.9

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right) \cdot x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot s\right)} \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot s\right) \cdot x\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(c \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)} \cdot x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(c \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)} \cdot x\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x\right)} \]

      14. lower-*.f64 71.3

          \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot x\right)}\right)\right) \cdot x\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \left(s \cdot \color{blue}{\left(x \cdot s\right)}\right)\right) \cdot x\right)} \]

      17. lower-*.f64 71.3

          \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot \left(s \cdot \color{blue}{\left(x \cdot s\right)}\right)\right) \cdot x\right)} \]

   9. Applied rewrites 71.3%

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right) \cdot x\right)}} \]

   if 5.4999999999999998e-163 < 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-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 98.0

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 98.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      13. lower-*.f64 67.0

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   7. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ (/ 1.0 t_0) t_0)))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return (1.0 / t_0) / t_0;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = (1.0d0 / t_0) / t_0
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return (1.0 / t_0) / t_0;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return (1.0 / t_0) / t_0

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 69.6

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Applied rewrites 69.6%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   13. swap-sqr N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

7. Applied rewrites 80.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

8. Final simplification 80.1%

   \[\leadsto \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)} \]
9. Add Preprocessing

Alternative 14: 79.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ 1.0 (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return 1.0 / (t_0 * t_0);
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return 1.0 / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return 1.0 / (t_0 * t_0)

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 69.6

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Applied rewrites 69.6%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   11. swap-sqr N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   14. lower-*.f64 79.3

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   17. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   19. lift-*.f64 78.4

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

   22. associate-*r* N/A

       \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

   23. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

   24. lift-*.f64 79.8

       \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

7. Applied rewrites 79.8%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

8. Final simplification 79.8%

   \[\leadsto \frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]
9. Add Preprocessing

Alternative 15: 78.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(\left(x \cdot s\_m\right) \cdot \left(c\_m \cdot \left(x \cdot s\_m\right)\right)\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* c_m (* (* x s_m) (* c_m (* x s_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * ((x * s_m) * (c_m * (x * s_m))))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))))

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(c_m * Float64(Float64(x * s_m) * Float64(c_m * Float64(x * s_m)))))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(c$95$m * N[(N[(x * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 69.6

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Applied rewrites 69.6%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right) \cdot x}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

   7. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot x} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x} \]

   10. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(s \cdot x\right) \cdot \left(\left(c \cdot \left(c \cdot s\right)\right) \cdot x\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot s\right)\right)} \cdot x\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(s \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

   14. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

   18. lower-*.f64 76.6

       \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

   21. associate-*r* N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   22. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

7. Applied rewrites 78.1%

   \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]

8. Final simplification 78.1%

   \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]
9. Add Preprocessing

Alternative 16: 67.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* c_m (* c_m (* s_m (* s_m (* x x)))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * (c_m * (s_m * (s_m * (x * x)))))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))))

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(s_m * Float64(x * x))))))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x * x)))));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

   10. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   12. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   15. lower-*.f64 96.9

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied rewrites 96.9%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   13. lower-*.f64 66.1

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

7. Applied rewrites 66.1%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))