mixedcos

Percentage Accurate: 66.2% → 99.1%

Time: 8.9s

Alternatives: 10

Speedup: 9.0×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% 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: 99.1% accurate, 0.6× speedup

Math

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

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 (cos (* x 2.0))) (t_1 (* (* s x) c_m)))
   (if (<= (/ t_0 (* (* (* (pow s 2.0) x) x) (pow c_m 2.0))) INFINITY)
     (/ (/ (cos (+ x x)) t_1) t_1)
     (/ 1.0 (/ (pow (* (* s c_m) x) 2.0) 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 = cos((x * 2.0));
	double t_1 = (s * x) * c_m;
	double tmp;
	if ((t_0 / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= ((double) INFINITY)) {
		tmp = (cos((x + x)) / t_1) / t_1;
	} else {
		tmp = 1.0 / (pow(((s * c_m) * x), 2.0) / t_0);
	}
	return tmp;
}

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 * x) * c_m;
	double tmp;
	if ((t_0 / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.cos((x + x)) / t_1) / t_1;
	} else {
		tmp = 1.0 / (Math.pow(((s * c_m) * x), 2.0) / 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 = math.cos((x * 2.0))
	t_1 = (s * x) * c_m
	tmp = 0
	if (t_0 / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= math.inf:
		tmp = (math.cos((x + x)) / t_1) / t_1
	else:
		tmp = 1.0 / (math.pow(((s * c_m) * x), 2.0) / 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 = cos(Float64(x * 2.0))
	t_1 = Float64(Float64(s * x) * c_m)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= Inf)
		tmp = Float64(Float64(cos(Float64(x + x)) / t_1) / t_1);
	else
		tmp = Float64(1.0 / Float64((Float64(Float64(s * c_m) * x) ^ 2.0) / 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 = cos((x * 2.0));
	t_1 = (s * x) * c_m;
	tmp = 0.0;
	if ((t_0 / ((((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= Inf)
		tmp = (cos((x + x)) / t_1) / t_1;
	else
		tmp = 1.0 / ((((s * c_m) * x) ^ 2.0) / 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[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c$95$m), $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], Infinity], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 / N[(N[Power[N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision] / 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))) < +inf.0

   1. Initial program 82.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if +inf.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 0.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 0.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 97.4

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 97.4

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

   4. Applied rewrites 97.4%

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

4. Final simplification 99.2%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

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

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 x) c_m)))
   (if (<= (/ t_0 (* (* (* (pow s 2.0) x) x) (pow c_m 2.0))) INFINITY)
     (/ (/ (cos (+ x x)) t_1) t_1)
     (/ t_0 (pow (* (* s c_m) x) 2.0)))))

C

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

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 * x) * c_m;
	double tmp;
	if ((t_0 / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.cos((x + x)) / t_1) / t_1;
	} else {
		tmp = t_0 / Math.pow(((s * c_m) * x), 2.0);
	}
	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 = math.cos((x * 2.0))
	t_1 = (s * x) * c_m
	tmp = 0
	if (t_0 / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= math.inf:
		tmp = (math.cos((x + x)) / t_1) / t_1
	else:
		tmp = t_0 / math.pow(((s * c_m) * x), 2.0)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(Float64(s * x) * c_m)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= Inf)
		tmp = Float64(Float64(cos(Float64(x + x)) / t_1) / t_1);
	else
		tmp = Float64(t_0 / (Float64(Float64(s * c_m) * x) ^ 2.0));
	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 = cos((x * 2.0));
	t_1 = (s * x) * c_m;
	tmp = 0.0;
	if ((t_0 / ((((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= Inf)
		tmp = (cos((x + x)) / t_1) / t_1;
	else
		tmp = t_0 / (((s * c_m) * x) ^ 2.0);
	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[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c$95$m), $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], Infinity], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / N[Power[N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision], 2.0], $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))) < +inf.0

   1. Initial program 82.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if +inf.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 0.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.4

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

   4. Applied rewrites 97.4%

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

4. Final simplification 99.2%

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

Alternative 3: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-2}{s}}{s}}{c\_m}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 x) c_m)))
   (if (<=
        (/ (cos (* x 2.0)) (* (* (* (pow s 2.0) x) x) (pow c_m 2.0)))
        -2e-69)
     (/ (/ (/ (/ -2.0 s) s) c_m) c_m)
     (/ (/ 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 * x) * c_m;
	double tmp;
	if ((cos((x * 2.0)) / (((pow(s, 2.0) * x) * x) * pow(c_m, 2.0))) <= -2e-69) {
		tmp = (((-2.0 / s) / s) / c_m) / c_m;
	} 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 = (s * x) * c_m
    if ((cos((x * 2.0d0)) / ((((s ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= (-2d-69)) then
        tmp = ((((-2.0d0) / s) / s) / c_m) / c_m
    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 = (s * x) * c_m;
	double tmp;
	if ((Math.cos((x * 2.0)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= -2e-69) {
		tmp = (((-2.0 / s) / s) / c_m) / c_m;
	} 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 = (s * x) * c_m
	tmp = 0
	if (math.cos((x * 2.0)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= -2e-69:
		tmp = (((-2.0 / s) / s) / c_m) / c_m
	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(s * x) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = Float64(Float64(Float64(Float64(-2.0 / s) / s) / c_m) / c_m);
	else
		tmp = Float64(Float64(1.0 / 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 * x) * c_m;
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((((s ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = (((-2.0 / s) / s) / c_m) / c_m;
	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[(s * x), $MachinePrecision] * c$95$m), $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], -2e-69], N[(N[(N[(N[(-2.0 / s), $MachinePrecision] / s), $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.9999999999999999e-69

   1. Initial program 51.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 37.9%

       \[\leadsto \frac{\frac{\frac{\frac{-2}{s}}{s}}{c}}{\color{blue}{c}} \]

   if -1.9999999999999999e-69 < (/.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 67.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-2}{s}}{s}}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \left(\left(\left(s \cdot c\_m\right) \cdot s\right) \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{{\left(c\_m \cdot x\right)}^{2} \cdot s}}{-s}\\ \end{array} \end{array} \]

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 x) c_m)))
   (if (<= x 1.1e-26)
     (/ (/ 1.0 t_0) t_0)
     (if (<= x 1.4e+150)
       (/ (cos (+ x x)) (* (* x x) (* (* (* s c_m) s) c_m)))
       (/ (/ -1.0 (* (pow (* c_m x) 2.0) s)) (- s))))))

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 * x) * c_m;
	double tmp;
	if (x <= 1.1e-26) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x <= 1.4e+150) {
		tmp = cos((x + x)) / ((x * x) * (((s * c_m) * s) * c_m));
	} else {
		tmp = (-1.0 / (pow((c_m * x), 2.0) * s)) / -s;
	}
	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 = (s * x) * c_m
    if (x <= 1.1d-26) then
        tmp = (1.0d0 / t_0) / t_0
    else if (x <= 1.4d+150) then
        tmp = cos((x + x)) / ((x * x) * (((s * c_m) * s) * c_m))
    else
        tmp = ((-1.0d0) / (((c_m * x) ** 2.0d0) * s)) / -s
    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 * x) * c_m;
	double tmp;
	if (x <= 1.1e-26) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x <= 1.4e+150) {
		tmp = Math.cos((x + x)) / ((x * x) * (((s * c_m) * s) * c_m));
	} else {
		tmp = (-1.0 / (Math.pow((c_m * x), 2.0) * s)) / -s;
	}
	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 * x) * c_m
	tmp = 0
	if x <= 1.1e-26:
		tmp = (1.0 / t_0) / t_0
	elif x <= 1.4e+150:
		tmp = math.cos((x + x)) / ((x * x) * (((s * c_m) * s) * c_m))
	else:
		tmp = (-1.0 / (math.pow((c_m * x), 2.0) * s)) / -s
	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 * x) * c_m)
	tmp = 0.0
	if (x <= 1.1e-26)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	elseif (x <= 1.4e+150)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(x * x) * Float64(Float64(Float64(s * c_m) * s) * c_m)));
	else
		tmp = Float64(Float64(-1.0 / Float64((Float64(c_m * x) ^ 2.0) * s)) / Float64(-s));
	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 * x) * c_m;
	tmp = 0.0;
	if (x <= 1.1e-26)
		tmp = (1.0 / t_0) / t_0;
	elseif (x <= 1.4e+150)
		tmp = cos((x + x)) / ((x * x) * (((s * c_m) * s) * c_m));
	else
		tmp = (-1.0 / (((c_m * x) ^ 2.0) * s)) / -s;
	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 * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 1.1e-26], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 1.4e+150], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Power[N[(c$95$m * x), $MachinePrecision], 2.0], $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision] / (-s)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.1e-26

   1. Initial program 62.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 80.8%

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

   if 1.1e-26 < x < 1.40000000000000005e150

   1. Initial program 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 89.7

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

   6. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 1.40000000000000005e150 < x

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 82.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot s\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{{\left(c \cdot x\right)}^{2} \cdot s}}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\\ t_1 := \left(s \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x) s)) (t_1 (* (* s x) c_m)))
   (if (<= x 3.1e+38)
     (/ (/ (cos (+ x x)) t_1) t_1)
     (/ (cos (* x 2.0)) (* 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 t_1 = (s * x) * c_m;
	double tmp;
	if (x <= 3.1e+38) {
		tmp = (cos((x + x)) / t_1) / t_1;
	} else {
		tmp = cos((x * 2.0)) / (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 = (c_m * x) * s
    t_1 = (s * x) * c_m
    if (x <= 3.1d+38) then
        tmp = (cos((x + x)) / t_1) / t_1
    else
        tmp = cos((x * 2.0d0)) / (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 t_1 = (s * x) * c_m;
	double tmp;
	if (x <= 3.1e+38) {
		tmp = (Math.cos((x + x)) / t_1) / t_1;
	} else {
		tmp = Math.cos((x * 2.0)) / (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
	t_1 = (s * x) * c_m
	tmp = 0
	if x <= 3.1e+38:
		tmp = (math.cos((x + x)) / t_1) / t_1
	else:
		tmp = math.cos((x * 2.0)) / (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)
	t_1 = Float64(Float64(s * x) * c_m)
	tmp = 0.0
	if (x <= 3.1e+38)
		tmp = Float64(Float64(cos(Float64(x + x)) / t_1) / t_1);
	else
		tmp = Float64(cos(Float64(x * 2.0)) / 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;
	t_1 = (s * x) * c_m;
	tmp = 0.0;
	if (x <= 3.1e+38)
		tmp = (cos((x + x)) / t_1) / t_1;
	else
		tmp = cos((x * 2.0)) / (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]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 3.1e+38], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000018e38

   1. Initial program 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 96.8

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

   8. Applied rewrites 96.8%

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

   if 3.10000000000000018e38 < x

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\\ t_1 := \left(s \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x) s)) (t_1 (* (* s x) c_m)))
   (if (<= x 5e-14) (/ (/ 1.0 t_1) t_1) (/ (cos (* x 2.0)) (* 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 t_1 = (s * x) * c_m;
	double tmp;
	if (x <= 5e-14) {
		tmp = (1.0 / t_1) / t_1;
	} else {
		tmp = cos((x * 2.0)) / (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 = (c_m * x) * s
    t_1 = (s * x) * c_m
    if (x <= 5d-14) then
        tmp = (1.0d0 / t_1) / t_1
    else
        tmp = cos((x * 2.0d0)) / (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 t_1 = (s * x) * c_m;
	double tmp;
	if (x <= 5e-14) {
		tmp = (1.0 / t_1) / t_1;
	} else {
		tmp = Math.cos((x * 2.0)) / (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
	t_1 = (s * x) * c_m
	tmp = 0
	if x <= 5e-14:
		tmp = (1.0 / t_1) / t_1
	else:
		tmp = math.cos((x * 2.0)) / (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)
	t_1 = Float64(Float64(s * x) * c_m)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	else
		tmp = Float64(cos(Float64(x * 2.0)) / 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;
	t_1 = (s * x) * c_m;
	tmp = 0.0;
	if (x <= 5e-14)
		tmp = (1.0 / t_1) / t_1;
	else
		tmp = cos((x * 2.0)) / (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]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 5e-14], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 62.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 96.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 81.1%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 7.8× speedup

Math

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

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 x) c_m))) (/ (/ 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 * x) * c_m;
	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 * x) * c_m
    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 * x) * c_m;
	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 * x) * c_m
	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 * x) * c_m)
	return Float64(Float64(1.0 / 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 * x) * c_m;
	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 * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 97.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. mul-1-neg N/A

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

   7. lift-neg.f64 N/A

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

   8. remove-double-neg N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

6. Applied rewrites 97.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 77.1%

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

Alternative 8: 78.5% accurate, 9.0× speedup

Math

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

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 x) c_m))) (/ 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 * x) * c_m;
	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 * x) * c_m
    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 * x) * c_m;
	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 * x) * c_m
	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 * x) * c_m)
	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 * x) * c_m;
	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 * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   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. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 79.4

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

4. Applied rewrites 79.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 75.9

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

6. Applied rewrites 75.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 65.0%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*l* N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. pow2 N/A

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

    9. pow2 N/A

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

    10. unpow-prod-down N/A

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

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. *-commutative N/A

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

    14. associate-*r* N/A

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

    15. *-commutative N/A

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

    16. lift-*.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. pow2 N/A

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

    19. lower-*.f64 77.0

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

    20. lift-*.f64 N/A

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

    21. *-commutative N/A

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

    22. lower-*.f64 77.0

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

    23. lift-*.f64 N/A

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

    24. *-commutative N/A

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

    25. lower-*.f64 77.0

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

11. Applied rewrites 77.0%

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

12. Final simplification 77.0%

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

Alternative 9: 69.5% accurate, 9.0× speedup

Math

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

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 c_m) s) x) 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 * c_m) * s) * x) * 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 * c_m) * s) * x) * 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 * c_m) * s) * x) * 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 * c_m) * s) * x) * 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(Float64(Float64(s * c_m) * s) * x) * 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 * c_m) * s) * x) * 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[(N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   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. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 79.4

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

4. Applied rewrites 79.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 75.9

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

6. Applied rewrites 75.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 65.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f64 N/A

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

    9. associate-*l* N/A

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

    10. lift-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-*.f64 70.2

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

11. Applied rewrites 70.2%

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

12. Final simplification 70.2%

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

Alternative 10: 68.4% accurate, 9.0× speedup

Math

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

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 x) x) c_m) s) 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 * x) * x) * c_m) * s) * 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 * x) * x) * c_m) * s) * 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 * x) * x) * c_m) * s) * 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 * x) * x) * c_m) * s) * 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(Float64(Float64(c_m * x) * x) * c_m) * s) * 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 * x) * x) * c_m) * s) * 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[(N[(N[(c$95$m * x), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. lift-*.f64 N/A

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

   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. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 79.4

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

4. Applied rewrites 79.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 75.9

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

6. Applied rewrites 75.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 65.0%

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

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. *-commutative N/A

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

    8. unpow2 N/A

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. associate-*r* N/A

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

    14. lower-*.f64 N/A

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

    15. lower-*.f64 69.6

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

12. Applied rewrites 69.6%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))