mixedcos

Percentage Accurate: 66.8% → 98.1%

Time: 8.0s

Alternatives: 13

Speedup: 6.2×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(-2 \cdot x\right)\\ t_1 := \left(c\_m \cdot s\_m\right) \cdot x\\ \mathbf{if}\;c\_m \leq 5.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{c\_m \cdot c\_m}}{s\_m \cdot x}}{s\_m \cdot x}\\ \end{array} \end{array} \]

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 s_m) x)))
   (if (<= c_m 5.2e-34)
     (/ (/ t_0 t_1) t_1)
     (/ (/ (/ t_0 (* c_m c_m)) (* s_m x)) (* s_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 * s_m) * x;
	double tmp;
	if (c_m <= 5.2e-34) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / (c_m * c_m)) / (s_m * x)) / (s_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 * s_m) * x
    if (c_m <= 5.2d-34) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = ((t_0 / (c_m * c_m)) / (s_m * x)) / (s_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 * s_m) * x;
	double tmp;
	if (c_m <= 5.2e-34) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / (c_m * c_m)) / (s_m * x)) / (s_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 * s_m) * x
	tmp = 0
	if c_m <= 5.2e-34:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = ((t_0 / (c_m * c_m)) / (s_m * x)) / (s_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(Float64(c_m * s_m) * x)
	tmp = 0.0
	if (c_m <= 5.2e-34)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(c_m * c_m)) / Float64(s_m * x)) / Float64(s_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 * s_m) * x;
	tmp = 0.0;
	if (c_m <= 5.2e-34)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = ((t_0 / (c_m * c_m)) / (s_m * x)) / (s_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[(N[(c$95$m * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[c$95$m, 5.2e-34], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 / N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 5.1999999999999999e-34

   1. Initial program 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 96.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 96.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-+PI/2-rev N/A

          \[\leadsto \frac{\frac{\color{blue}{\sin \left(2 \cdot x + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot c\right) \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\sin \left(\color{blue}{2 \cdot x} + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot c\right) \cdot x} \]

      13. metadata-eval N/A

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

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

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

      15. sin-+PI/2-rev N/A

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

      16. cos-neg-rev N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   if 5.1999999999999999e-34 < c

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 91.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 91.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 91.9

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. metadata-eval N/A

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

      6. cos-neg-rev N/A

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

      7. cos-neg-rev N/A

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

      8. metadata-eval N/A

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

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

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

      10. cos-neg-rev N/A

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

      11. cos-neg-rev N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. unswap-sqr N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   9. Applied rewrites 89.7%

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

Alternative 2: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{-1} - 2}{\left(\left(s\_m \cdot c\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 (<=
      (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x)))
      -5e-227)
   (/ (- (pow (* x x) -1.0) 2.0) (* (* (* s_m c_m) c_m) s_m))
   (/ 1.0 (* (* (* x (* s_m c_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 tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -5e-227) {
		tmp = (pow((x * x), -1.0) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / (((x * (s_m * c_m)) * (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) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-5d-227)) then
        tmp = (((x * x) ** (-1.0d0)) - 2.0d0) / (((s_m * c_m) * c_m) * s_m)
    else
        tmp = 1.0d0 / (((x * (s_m * c_m)) * (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 tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -5e-227) {
		tmp = (Math.pow((x * x), -1.0) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / (((x * (s_m * c_m)) * (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):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -5e-227:
		tmp = (math.pow((x * x), -1.0) - 2.0) / (((s_m * c_m) * c_m) * s_m)
	else:
		tmp = 1.0 / (((x * (s_m * c_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)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -5e-227)
		tmp = Float64(Float64((Float64(x * x) ^ -1.0) - 2.0) / Float64(Float64(Float64(s_m * c_m) * c_m) * s_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -5e-227)
		tmp = (((x * x) ^ -1.0) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	else
		tmp = 1.0 / (((x * (s_m * c_m)) * (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_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-227], N[(N[(N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $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))) < -4.99999999999999961e-227

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

   7. Applied rewrites 40.4%

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

   if -4.99999999999999961e-227 < (/.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 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 95.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 95.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 83.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{-1} - 2}{\left(\left(s \cdot c\right) \cdot c\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\frac{2}{\left(\left(\left(-c\_m\right) \cdot s\_m\right) \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 (<=
      (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x)))
      -5e-227)
   (/ 2.0 (* (* (* (- c_m) s_m) s_m) c_m))
   (/ 1.0 (* (* (* x (* s_m c_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 tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -5e-227) {
		tmp = 2.0 / (((-c_m * s_m) * s_m) * c_m);
	} else {
		tmp = 1.0 / (((x * (s_m * c_m)) * (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) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-5d-227)) then
        tmp = 2.0d0 / (((-c_m * s_m) * s_m) * c_m)
    else
        tmp = 1.0d0 / (((x * (s_m * c_m)) * (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 tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -5e-227) {
		tmp = 2.0 / (((-c_m * s_m) * s_m) * c_m);
	} else {
		tmp = 1.0 / (((x * (s_m * c_m)) * (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):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -5e-227:
		tmp = 2.0 / (((-c_m * s_m) * s_m) * c_m)
	else:
		tmp = 1.0 / (((x * (s_m * c_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)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -5e-227)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(-c_m) * s_m) * s_m) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -5e-227)
		tmp = 2.0 / (((-c_m * s_m) * s_m) * c_m);
	else
		tmp = 1.0 / (((x * (s_m * c_m)) * (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_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-227], N[(2.0 / N[(N[(N[((-c$95$m) * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $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))) < -4.99999999999999961e-227

   1. Initial program 79.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.4%

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

   10. Applied rewrites 40.4%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 40.4%

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

   if -4.99999999999999961e-227 < (/.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 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 95.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 95.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 83.8%

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

Alternative 4: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(c\_m \cdot x\right) \cdot s\_m\right)}^{2}\right)}^{-1}\\ \end{array} \end{array} \]

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 (* (* s_m x) c_m)))
   (if (<= x 7.2e+42)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (pow (pow (* (* c_m x) s_m) 2.0) -1.0))))

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 = (s_m * x) * c_m;
	double tmp;
	if (x <= 7.2e+42) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = pow(pow(((c_m * x) * s_m), 2.0), -1.0);
	}
	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(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 7.2e+42)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = (Float64(Float64(c_m * x) * s_m) ^ 2.0) ^ -1.0;
	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[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 7.2e+42], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(c$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000002e42

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 75.0%

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

   if 7.2000000000000002e42 < x

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   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. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{{s}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}}{s} \]

      8. unpow2 N/A

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

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{1}{{c}^{2}}}{x}}}{x}}{s}}{s} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{1}{{c}^{2}}}}{x}}{x}}{s}}{s} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

      14. lower-*.f64 61.7

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

   7. Applied rewrites 61.7%

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

   9. Applied rewrites 62.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(c \cdot x\right) \cdot s\right)}^{2}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\_m\right) \cdot x\\ \mathbf{if}\;c\_m \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\cos \left(-2 \cdot x\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\right) \cdot \left(s\_m \cdot x\right)\right)}^{-1}}{s\_m}\\ \end{array} \end{array} \]

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 s_m) x)))
   (if (<= c_m 1.7e+62)
     (/ (/ (cos (* -2.0 x)) t_0) t_0)
     (/ (pow (* (* (* c_m c_m) x) (* s_m x)) -1.0) 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 * s_m) * x;
	double tmp;
	if (c_m <= 1.7e+62) {
		tmp = (cos((-2.0 * x)) / t_0) / t_0;
	} else {
		tmp = pow((((c_m * c_m) * x) * (s_m * x)), -1.0) / 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 * s_m) * x
    if (c_m <= 1.7d+62) then
        tmp = (cos(((-2.0d0) * x)) / t_0) / t_0
    else
        tmp = ((((c_m * c_m) * x) * (s_m * x)) ** (-1.0d0)) / 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 * s_m) * x;
	double tmp;
	if (c_m <= 1.7e+62) {
		tmp = (Math.cos((-2.0 * x)) / t_0) / t_0;
	} else {
		tmp = Math.pow((((c_m * c_m) * x) * (s_m * x)), -1.0) / 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 * s_m) * x
	tmp = 0
	if c_m <= 1.7e+62:
		tmp = (math.cos((-2.0 * x)) / t_0) / t_0
	else:
		tmp = math.pow((((c_m * c_m) * x) * (s_m * x)), -1.0) / 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(Float64(c_m * s_m) * x)
	tmp = 0.0
	if (c_m <= 1.7e+62)
		tmp = Float64(Float64(cos(Float64(-2.0 * x)) / t_0) / t_0);
	else
		tmp = Float64((Float64(Float64(Float64(c_m * c_m) * x) * Float64(s_m * x)) ^ -1.0) / 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 * s_m) * x;
	tmp = 0.0;
	if (c_m <= 1.7e+62)
		tmp = (cos((-2.0 * x)) / t_0) / t_0;
	else
		tmp = ((((c_m * c_m) * x) * (s_m * x)) ^ -1.0) / 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[(N[(c$95$m * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[c$95$m, 1.7e+62], N[(N[(N[Cos[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.70000000000000007e62

   1. Initial program 69.4%

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

      1. lift-*.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-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 96.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 96.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-+PI/2-rev N/A

          \[\leadsto \frac{\frac{\color{blue}{\sin \left(2 \cdot x + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot c\right) \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\sin \left(\color{blue}{2 \cdot x} + \frac{\mathsf{PI}\left(\right)}{2}\right)}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot c\right) \cdot x} \]

      13. metadata-eval N/A

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

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

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

      15. sin-+PI/2-rev N/A

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

      16. cos-neg-rev N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 98.1%

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

   if 1.70000000000000007e62 < c

   1. Initial program 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   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. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{{s}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}}{s} \]

      8. unpow2 N/A

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

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{1}{{c}^{2}}}{x}}}{x}}{s}}{s} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{1}{{c}^{2}}}}{x}}{x}}{s}}{s} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

      14. lower-*.f64 78.5

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 80.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\cos \left(-2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right)}^{-1}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 (* (* s_m x) c_m)))
   (if (<= x 2e-123)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ (cos (+ x x)) (* (* (* x (* s_m c_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 = (s_m * x) * c_m;
	double tmp;
	if (x <= 2e-123) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / (((x * (s_m * c_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 = Float64(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 2e-123)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 2e-123], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-123

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 53.5%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 71.1%

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

   if 2.0000000000000001e-123 < x

   1. Initial program 69.9%

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

      1. lift-*.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-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 98.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 98.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 98.0

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

   8. Applied rewrites 98.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 1.04 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot c\_m\right) \cdot \left(\left(\left(s\_m \cdot s\_m\right) \cdot x\right) \cdot c\_m\right)}\\ \end{array} \end{array} \]

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 (* (* s_m x) c_m)))
   (if (<= x 1.04e-23)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ (cos (+ x x)) (* (* x c_m) (* (* (* s_m s_m) x) c_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 = (s_m * x) * c_m;
	double tmp;
	if (x <= 1.04e-23) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / ((x * c_m) * (((s_m * s_m) * x) * c_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(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 1.04e-23)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(x * c_m) * Float64(Float64(Float64(s_m * s_m) * x) * c_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[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 1.04e-23], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * c$95$m), $MachinePrecision] * N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e-23

   1. Initial program 70.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 72.5%

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

   9. Applied rewrites 74.3%

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

   if 1.04e-23 < x

   1. Initial program 67.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 82.4

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 82.4

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

   6. Applied rewrites 82.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.04 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left(\left(\left(s \cdot s\right) \cdot x\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(c\_m \cdot x\right) \cdot s\_m\right)}^{-2}\\ \end{array} \end{array} \]

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 (* (* s_m x) c_m)))
   (if (<= x 7.2e+42)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (pow (* (* c_m x) s_m) -2.0))))

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 = (s_m * x) * c_m;
	double tmp;
	if (x <= 7.2e+42) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = pow(((c_m * x) * s_m), -2.0);
	}
	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(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 7.2e+42)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(c_m * x) * s_m) ^ -2.0;
	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[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 7.2e+42], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision], -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000002e42

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 75.0%

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

   if 7.2000000000000002e42 < x

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      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. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   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. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{{s}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}{s}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{{x}^{2}}}{s}}}{s} \]

      8. unpow2 N/A

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

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{1}{{c}^{2}}}{x}}{x}}}{s}}{s} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{1}{{c}^{2}}}{x}}}{x}}{s}}{s} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{1}{{c}^{2}}}}{x}}{x}}{s}}{s} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

      14. lower-*.f64 61.7

          \[\leadsto \frac{\frac{\frac{\frac{\frac{1}{\color{blue}{c \cdot c}}}{x}}{x}}{s}}{s} \]

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{1}{c \cdot c}}{x}}{x}}{s}}{s}} \]

   9. Applied rewrites 62.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(c \cdot x\right) \cdot s\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 (* (* s_m x) c_m)))
   (if (<= x 7.2e+42)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ 1.0 (* (* (* x (* s_m c_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 = (s_m * x) * c_m;
	double tmp;
	if (x <= 7.2e+42) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((x * (s_m * c_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 = Float64(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 7.2e+42)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 7.2e+42], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000002e42

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 75.0%

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

   if 7.2000000000000002e42 < x

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 97.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot c\_m\right) \cdot x\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 (* (* s_m c_m) x)))
   (if (<= x 7.2e+42)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ 1.0 (* (* (* x (* s_m c_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 = (s_m * c_m) * x;
	double tmp;
	if (x <= 7.2e+42) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((x * (s_m * c_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 = Float64(Float64(s_m * c_m) * x)
	tmp = 0.0
	if (x <= 7.2e+42)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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[(N[(s$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 7.2e+42], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000002e42

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 73.2%

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

   if 7.2000000000000002e42 < x

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 97.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.4%

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

Alternative 11: 69.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\_m\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x}\\ \end{array} \end{array} \]

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 7.2e+42)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ 1.0 (* (* (* x (* s_m c_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 = (c_m * x) * s_m;
	double tmp;
	if (x <= 7.2e+42) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((x * (s_m * c_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 = Float64(Float64(c_m * x) * s_m)
	tmp = 0.0
	if (x <= 7.2e+42)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * Float64(s_m * c_m)) * Float64(s_m * 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[(N[(c$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x, 7.2e+42], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000002e42

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 73.6%

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

   if 7.2000000000000002e42 < x

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 97.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.4%

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

Alternative 12: 31.9% accurate, 11.5× speedup

Math

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

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
 (/ 2.0 (* (* (* (- s_m) s_m) c_m) c_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 2.0 / (((-s_m * s_m) * c_m) * c_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 = 2.0d0 / (((-s_m * s_m) * c_m) * c_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 2.0 / (((-s_m * s_m) * c_m) * c_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 2.0 / (((-s_m * s_m) * c_m) * c_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(2.0 / Float64(Float64(Float64(Float64(-s_m) * s_m) * c_m) * c_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 = 2.0 / (((-s_m * s_m) * c_m) * c_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[(2.0 / N[(N[(N[((-s$95$m) * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.5%

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

10. Applied rewrites 28.8%

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

12. Applied rewrites 29.5%

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

Alternative 13: 28.7% accurate, 11.5× speedup

Math

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

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
 (/ 2.0 (* (* (* (- c_m) s_m) s_m) c_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 2.0 / (((-c_m * s_m) * s_m) * c_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 = 2.0d0 / (((-c_m * s_m) * s_m) * c_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 2.0 / (((-c_m * s_m) * s_m) * c_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 2.0 / (((-c_m * s_m) * s_m) * c_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(2.0 / Float64(Float64(Float64(Float64(-c_m) * s_m) * s_m) * c_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 = 2.0 / (((-c_m * s_m) * s_m) * c_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[(2.0 / N[(N[(N[((-c$95$m) * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.5%

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

10. Applied rewrites 28.8%

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

11. Taylor expanded in c around 0

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

13. Applied rewrites 28.5%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))