mixedcos

Percentage Accurate: 67.2% → 98.6%

Time: 11.5s

Alternatives: 12

Speedup: 9.0×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot c\_m\right) \cdot x\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;\frac{t\_1}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq 2 \cdot 10^{+206}:\\ \;\;\;\;{\left(\left(\left(-x\right) \cdot s\right) \cdot c\_m\right)}^{-2} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\cos x}^{4} - {\sin x}^{4}}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* s c_m) x)) (t_1 (cos (* x 2.0))))
   (if (<= (/ t_1 (* (* (* (pow s 2.0) x) x) (pow c_m 2.0))) 2e+206)
     (* (pow (* (* (- x) s) c_m) -2.0) t_1)
     (/ (- (pow (cos x) 4.0) (pow (sin x) 4.0)) (* t_0 t_0)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (s * c_m) * x;
	double t_1 = cos((x * 2.0));
	double tmp;
	if ((t_1 / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= 2e+206) {
		tmp = pow(((-x * s) * c_m), -2.0) * t_1;
	} else {
		tmp = (pow(cos(x), 4.0) - pow(sin(x), 4.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (s * c_m) * x
    t_1 = cos((x * 2.0d0))
    if ((t_1 / ((((s ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= 2d+206) then
        tmp = (((-x * s) * c_m) ** (-2.0d0)) * t_1
    else
        tmp = ((cos(x) ** 4.0d0) - (sin(x) ** 4.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = (s * c_m) * x;
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if ((t_1 / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= 2e+206) {
		tmp = Math.pow(((-x * s) * c_m), -2.0) * t_1;
	} else {
		tmp = (Math.pow(Math.cos(x), 4.0) - Math.pow(Math.sin(x), 4.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = (s * c_m) * x
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if (t_1 / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= 2e+206:
		tmp = math.pow(((-x * s) * c_m), -2.0) * t_1
	else:
		tmp = (math.pow(math.cos(x), 4.0) - math.pow(math.sin(x), 4.0)) / (t_0 * t_0)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(s * c_m) * x)
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= 2e+206)
		tmp = Float64((Float64(Float64(Float64(-x) * s) * c_m) ^ -2.0) * t_1);
	else
		tmp = Float64(Float64((cos(x) ^ 4.0) - (sin(x) ^ 4.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	t_0 = (s * c_m) * x;
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if ((t_1 / ((((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= 2e+206)
		tmp = (((-x * s) * c_m) ^ -2.0) * t_1;
	else
		tmp = ((cos(x) ^ 4.0) - (sin(x) ^ 4.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+206], N[(N[Power[N[(N[((-x) * s), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision] - N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $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))) < 2.0000000000000001e206

   1. Initial program 75.2%

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

      1. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. unswap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.6

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

   4. Applied rewrites 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 91.3%

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

   8. Applied rewrites 99.5%

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

   if 2.0000000000000001e206 < (/.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 49.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. unswap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. cos-sin-sum N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. cos-2 N/A

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

      7. difference-of-squares N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{{\cos x}^{\color{blue}{4}} - \left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{{\cos x}^{\color{blue}{4}} - \left(\sin x \cdot \sin x\right) \cdot \left(\sin x \cdot \sin x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      17. pow2 N/A

          \[\leadsto \frac{{\cos x}^{4} - \color{blue}{{\sin x}^{2}} \cdot \left(\sin x \cdot \sin x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      18. pow2 N/A

          \[\leadsto \frac{{\cos x}^{4} - {\sin x}^{2} \cdot \color{blue}{{\sin x}^{2}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      19. pow-prod-up N/A

          \[\leadsto \frac{{\cos x}^{4} - \color{blue}{{\sin x}^{\left(2 + 2\right)}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      20. metadata-eval N/A

          \[\leadsto \frac{{\cos x}^{4} - {\sin x}^{\color{blue}{4}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      21. metadata-eval N/A

          \[\leadsto \frac{{\cos x}^{4} - {\sin x}^{\color{blue}{\left(-2 \cdot -2\right)}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      22. lower-pow.f64 N/A

          \[\leadsto \frac{{\cos x}^{4} - \color{blue}{{\sin x}^{\left(-2 \cdot -2\right)}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      23. lower-sin.f64 N/A

          \[\leadsto \frac{{\cos x}^{4} - {\color{blue}{\sin x}}^{\left(-2 \cdot -2\right)}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

      24. metadata-eval 98.3

          \[\leadsto \frac{{\cos x}^{4} - {\sin x}^{\color{blue}{4}}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   6. Applied rewrites 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq 2 \cdot 10^{+206}:\\ \;\;\;\;{\left(\left(\left(-x\right) \cdot s\right) \cdot c\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\cos x}^{4} - {\sin x}^{4}}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := \left(s \cdot c\_m\right) \cdot x\\ \mathbf{if}\;\frac{t\_0}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq 0:\\ \;\;\;\;{\left(\left(\left(-x\right) \cdot s\right) \cdot c\_m\right)}^{-2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* (* s c_m) x)))
   (if (<= (/ t_0 (* (* (* (pow s 2.0) x) x) (pow c_m 2.0))) 0.0)
     (* (pow (* (* (- x) s) c_m) -2.0) t_0)
     (/ (/ (cos (+ x x)) t_1) t_1))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = (s * c_m) * x;
	double tmp;
	if ((t_0 / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= 0.0) {
		tmp = pow(((-x * s) * c_m), -2.0) * t_0;
	} else {
		tmp = (cos((x + x)) / t_1) / t_1;
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = (s * c_m) * x
    if ((t_0 / ((((s ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= 0.0d0) then
        tmp = (((-x * s) * c_m) ** (-2.0d0)) * t_0
    else
        tmp = (cos((x + x)) / t_1) / t_1
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = (s * c_m) * x;
	double tmp;
	if ((t_0 / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= 0.0) {
		tmp = Math.pow(((-x * s) * c_m), -2.0) * t_0;
	} else {
		tmp = (Math.cos((x + x)) / t_1) / t_1;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = math.cos((x * 2.0))
	t_1 = (s * c_m) * x
	tmp = 0
	if (t_0 / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= 0.0:
		tmp = math.pow(((-x * s) * c_m), -2.0) * t_0
	else:
		tmp = (math.cos((x + x)) / t_1) / t_1
	return tmp

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[N[(N[((-x) * s), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 0.0

   1. Initial program 71.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. unswap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 90.2%

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

   8. Applied rewrites 99.6%

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

   if 0.0 < (/.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 55.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. unswap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 97.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 97.4

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq 0:\\ \;\;\;\;{\left(\left(\left(-x\right) \cdot s\right) \cdot c\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot c\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot c\_m\right) \cdot x\\ t_1 := \left(c\_m \cdot x\right) \cdot s\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* s c_m) x)) (t_1 (* (* c_m x) s)))
   (if (<=
        (/ (cos (* x 2.0)) (* (* (* (pow s 2.0) x) x) (pow c_m 2.0)))
        -1e-94)
     (/ (fma (* x x) -2.0 1.0) (* t_0 t_0))
     (/ 1.0 (* t_1 t_1)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (s * c_m) * x;
	double t_1 = (c_m * x) * s;
	double tmp;
	if ((cos((x * 2.0)) / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= -1e-94) {
		tmp = fma((x * x), -2.0, 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (t_1 * t_1);
	}
	return tmp;
}

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(s * c_m) * x)
	t_1 = Float64(Float64(c_m * x) * s)
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -1e-94)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-94], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $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))) < -9.9999999999999996e-95

   1. Initial program 59.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.3

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

   5. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 41.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 41.3

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 41.3

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 41.3

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

      26. lift-*.f64 N/A

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

      27. *-commutative N/A

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

      28. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if -9.9999999999999996e-95 < (/.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 63.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 82.2

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

      17. remove-double-neg N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-neg-in N/A

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

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

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 82.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(s \cdot c\_m\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* c_m x) s)))
   (if (<=
        (/ (cos (* x 2.0)) (* (* (* (pow s 2.0) x) x) (pow c_m 2.0)))
        -1e+154)
     (/ 1.0 (* t_0 (* (* s c_m) x)))
     (/ 1.0 (* t_0 t_0)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (c_m * x) * s;
	double tmp;
	if ((cos((x * 2.0)) / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= -1e+154) {
		tmp = 1.0 / (t_0 * ((s * c_m) * x));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c_m * x) * s
    if ((cos((x * 2.0d0)) / ((((s ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= (-1d+154)) then
        tmp = 1.0d0 / (t_0 * ((s * c_m) * x))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = (c_m * x) * s;
	double tmp;
	if ((Math.cos((x * 2.0)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= -1e+154) {
		tmp = 1.0 / (t_0 * ((s * c_m) * x));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = (c_m * x) * s
	tmp = 0
	if (math.cos((x * 2.0)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= -1e+154:
		tmp = 1.0 / (t_0 * ((s * c_m) * x))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(c_m * x) * s)
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -1e+154)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(s * c_m) * x)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	t_0 = (c_m * x) * s;
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -1e+154)
		tmp = 1.0 / (t_0 * ((s * c_m) * x));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(c$95$m * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+154], N[(1.0 / N[(t$95$0 * N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $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))) < -1.00000000000000004e154

   1. Initial program 52.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 0.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 1.0

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

      17. remove-double-neg N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-neg-in N/A

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

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

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 1.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

      11. lower--.f64 23.8

          \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(0 - \color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\left(0 - \color{blue}{\left(s \cdot c\right)} \cdot x\right) \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

      14. lift-*.f64 23.8

          \[\leadsto \frac{1}{\left(0 - \color{blue}{\left(s \cdot c\right)} \cdot x\right) \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

   9. Applied rewrites 23.8%

      \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(s \cdot c\right) \cdot x\right)} \cdot \left(\left(-s\right) \cdot \left(c \cdot x\right)\right)} \]

   if -1.00000000000000004e154 < (/.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 63.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 81.2

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

      17. remove-double-neg N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-neg-in N/A

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

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

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 81.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot c\_m\right) \cdot x}}{\left(s \cdot x\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(c\_m \cdot x\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= x 6.2e-53)
   (/ (/ 1.0 (* (* s c_m) x)) (* (* s x) c_m))
   (/ (cos (+ x x)) (* (* (* (* c_m x) x) s) (* s c_m)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 6.2e-53) {
		tmp = (1.0 / ((s * c_m) * x)) / ((s * x) * c_m);
	} else {
		tmp = cos((x + x)) / ((((c_m * x) * x) * s) * (s * c_m));
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 6.2d-53) then
        tmp = (1.0d0 / ((s * c_m) * x)) / ((s * x) * c_m)
    else
        tmp = cos((x + x)) / ((((c_m * x) * x) * s) * (s * c_m))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 6.2e-53) {
		tmp = (1.0 / ((s * c_m) * x)) / ((s * x) * c_m);
	} else {
		tmp = Math.cos((x + x)) / ((((c_m * x) * x) * s) * (s * c_m));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if x <= 6.2e-53:
		tmp = (1.0 / ((s * c_m) * x)) / ((s * x) * c_m)
	else:
		tmp = math.cos((x + x)) / ((((c_m * x) * x) * s) * (s * c_m))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (x <= 6.2e-53)
		tmp = Float64(Float64(1.0 / Float64(Float64(s * c_m) * x)) / Float64(Float64(s * x) * c_m));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(Float64(c_m * x) * x) * s) * Float64(s * c_m)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if (x <= 6.2e-53)
		tmp = (1.0 / ((s * c_m) * x)) / ((s * x) * c_m);
	else
		tmp = cos((x + x)) / ((((c_m * x) * x) * s) * (s * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[x, 6.2e-53], N[(N[(1.0 / N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000031e-53

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 81.9

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

      17. remove-double-neg N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-neg-in N/A

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

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

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 81.3%

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

   if 6.20000000000000031e-53 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot c\right) \cdot x}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow2 N/A

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

   12. unswap-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 96.3

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

4. Applied rewrites 96.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 96.6

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 96.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 96.6

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 96.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 96.6

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 96.6

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

6. Applied rewrites 96.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

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

   4. lower-+.f64 96.6

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

8. Applied rewrites 96.6%

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

9. Final simplification 96.6%

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

Alternative 7: 75.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;{c\_m}^{2} \leq 0.001:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(s \cdot c\_m\right) \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot s\right) \cdot \left(\left(c\_m \cdot x\right) \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (pow c_m 2.0) 0.001)
   (/ 1.0 (* (* (* s x) x) (* (* s c_m) c_m)))
   (/ 1.0 (* (* (* s x) s) (* (* c_m x) c_m)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (pow(c_m, 2.0) <= 0.001) {
		tmp = 1.0 / (((s * x) * x) * ((s * c_m) * c_m));
	} else {
		tmp = 1.0 / (((s * x) * s) * ((c_m * x) * c_m));
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((c_m ** 2.0d0) <= 0.001d0) then
        tmp = 1.0d0 / (((s * x) * x) * ((s * c_m) * c_m))
    else
        tmp = 1.0d0 / (((s * x) * s) * ((c_m * x) * c_m))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (Math.pow(c_m, 2.0) <= 0.001) {
		tmp = 1.0 / (((s * x) * x) * ((s * c_m) * c_m));
	} else {
		tmp = 1.0 / (((s * x) * s) * ((c_m * x) * c_m));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if math.pow(c_m, 2.0) <= 0.001:
		tmp = 1.0 / (((s * x) * x) * ((s * c_m) * c_m))
	else:
		tmp = 1.0 / (((s * x) * s) * ((c_m * x) * c_m))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if ((c_m ^ 2.0) <= 0.001)
		tmp = Float64(1.0 / Float64(Float64(Float64(s * x) * x) * Float64(Float64(s * c_m) * c_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s * x) * s) * Float64(Float64(c_m * x) * c_m)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if ((c_m ^ 2.0) <= 0.001)
		tmp = 1.0 / (((s * x) * x) * ((s * c_m) * c_m));
	else
		tmp = 1.0 / (((s * x) * s) * ((c_m * x) * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[Power[c$95$m, 2.0], $MachinePrecision], 0.001], N[(1.0 / N[(N[(N[(s * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(s * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s * x), $MachinePrecision] * s), $MachinePrecision] * N[(N[(c$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 1e-3

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 61.0

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

   8. Applied rewrites 61.0%

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 69.5

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

   13. Applied rewrites 69.5%

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

   if 1e-3 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 60.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

   10. Applied rewrites 72.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 0.001:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow2 N/A

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

   12. unswap-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 96.3

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

4. Applied rewrites 96.3%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 96.3

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

6. Applied rewrites 96.3%

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

7. Final simplification 96.3%

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

Alternative 9: 78.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. unpow-prod-down N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 75.6

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

   17. remove-double-neg N/A

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

   18. lift-*.f64 N/A

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

   19. distribute-rgt-neg-in N/A

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

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

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

   21. lower-*.f64 N/A

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

7. Applied rewrites 75.9%

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

8. Final simplification 75.9%

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

Alternative 10: 78.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. unpow-prod-down N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 75.6

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 75.6

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 75.6

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 75.6

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

   26. lift-*.f64 N/A

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

   27. *-commutative N/A

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

   28. lower-*.f64 75.6

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

7. Applied rewrites 75.6%

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

8. Final simplification 75.6%

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

Alternative 11: 70.1% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 63.4

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

8. Applied rewrites 63.4%

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

10. Applied rewrites 67.3%

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

11. Final simplification 67.3%

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

Alternative 12: 65.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 63.4

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

8. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))