mixedcos

Percentage Accurate: 66.7% → 97.8%

Time: 7.9s

Alternatives: 11

Speedup: 9.0×

• 31.7% 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: 66.7% 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: 97.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{{\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<= x_m 1.5e-54)
     (/ (/ 1.0 c_m) (* (pow (* s_m x_m) 2.0) c_m))
     (/ (cos (+ x_m x_m)) (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if (x_m <= 1.5e-54) {
		tmp = (1.0 / c_m) / (pow((s_m * x_m), 2.0) * c_m);
	} else {
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (c_m * x_m)
    if (x_m <= 1.5d-54) then
        tmp = (1.0d0 / c_m) / (((s_m * x_m) ** 2.0d0) * c_m)
    else
        tmp = cos((x_m + x_m)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if (x_m <= 1.5e-54) {
		tmp = (1.0 / c_m) / (Math.pow((s_m * x_m), 2.0) * c_m);
	} else {
		tmp = Math.cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	tmp = 0
	if x_m <= 1.5e-54:
		tmp = (1.0 / c_m) / (math.pow((s_m * x_m), 2.0) * c_m)
	else:
		tmp = math.cos((x_m + x_m)) / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (x_m <= 1.5e-54)
		tmp = Float64(Float64(1.0 / c_m) / Float64((Float64(s_m * x_m) ^ 2.0) * c_m));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	tmp = 0.0;
	if (x_m <= 1.5e-54)
		tmp = (1.0 / c_m) / (((s_m * x_m) ^ 2.0) * c_m);
	else
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.5e-54], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000005e-54

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.1

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 73.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.9

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

   9. Applied rewrites 73.9%

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

   if 1.50000000000000005e-54 < x

   1. Initial program 71.9%

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

   3. Taylor expanded in x around 0

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

      1. *-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 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{c}}{{\left(s \cdot x\right)}^{2} \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{\left({\left(c\_m \cdot x\_m\right)}^{2} \cdot s\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{{\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
      -4e-116)
   (/ (fma -2.0 (* x_m x_m) 1.0) (* (* (pow (* c_m x_m) 2.0) s_m) s_m))
   (/ (/ 1.0 c_m) (* (pow (* s_m x_m) 2.0) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-116) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / ((pow((c_m * x_m), 2.0) * s_m) * s_m);
	} else {
		tmp = (1.0 / c_m) / (pow((s_m * x_m), 2.0) * c_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(Float64((Float64(c_m * x_m) ^ 2.0) * s_m) * s_m));
	else
		tmp = Float64(Float64(1.0 / c_m) / Float64((Float64(s_m * x_m) ^ 2.0) * c_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-116], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(c$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $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))) < -4e-116

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.0

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

   7. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 38.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 38.2

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

   9. Applied rewrites 38.2%

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

   if -4e-116 < (/.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 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.3

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 79.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.4

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

   9. Applied rewrites 79.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left({\left(c \cdot x\right)}^{2} \cdot s\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{{\left(s \cdot x\right)}^{2} \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{{\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-116)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ 1.0 c_m) (* (pow (* s_m x_m) 2.0) c_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-116) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / c_m) / (pow((s_m * x_m), 2.0) * c_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / c_m) / Float64((Float64(s_m * x_m) ^ 2.0) * c_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-116], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $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))) < -4e-116

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -4e-116 < (/.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 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.3

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 79.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.4

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

   9. Applied rewrites 79.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{{\left(s \cdot x\right)}^{2} \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s\_m \cdot x\_m\right)}^{-2}}{c\_m}}{c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-116)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ (pow (* s_m x_m) -2.0) c_m) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-116) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (pow((s_m * x_m), -2.0) / c_m) / c_m;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64((Float64(s_m * x_m) ^ -2.0) / c_m) / c_m);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-116], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], -2.0], $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $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))) < -4e-116

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -4e-116 < (/.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 67.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 79.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot x\right)}^{-2}}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-116)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-116) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-116], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4e-116

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -4e-116 < (/.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 67.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(c \cdot x\right)}}{s \cdot \left(c \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m\right) \cdot -2}{\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-116)
     (/ (* (* x_m x_m) -2.0) (* (* (* (* c_m c_m) x_m) (* s_m x_m)) s_m))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-116) {
		tmp = ((x_m * x_m) * -2.0) / ((((c_m * c_m) * x_m) * (s_m * x_m)) * s_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (c_m * x_m)
    if ((cos((2.0d0 * x_m)) / ((((s_m ** 2.0d0) * x_m) * x_m) * (c_m ** 2.0d0))) <= (-4d-116)) then
        tmp = ((x_m * x_m) * (-2.0d0)) / ((((c_m * c_m) * x_m) * (s_m * x_m)) * s_m)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (((Math.pow(s_m, 2.0) * x_m) * x_m) * Math.pow(c_m, 2.0))) <= -4e-116) {
		tmp = ((x_m * x_m) * -2.0) / ((((c_m * c_m) * x_m) * (s_m * x_m)) * s_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	tmp = 0
	if (math.cos((2.0 * x_m)) / (((math.pow(s_m, 2.0) * x_m) * x_m) * math.pow(c_m, 2.0))) <= -4e-116:
		tmp = ((x_m * x_m) * -2.0) / ((((c_m * c_m) * x_m) * (s_m * x_m)) * s_m)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = Float64(Float64(Float64(x_m * x_m) * -2.0) / Float64(Float64(Float64(Float64(c_m * c_m) * x_m) * Float64(s_m * x_m)) * s_m));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-116)
		tmp = ((x_m * x_m) * -2.0) / ((((c_m * c_m) * x_m) * (s_m * x_m)) * s_m);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-116], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4e-116

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.0

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

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 39.0%

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

   if -4e-116 < (/.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 67.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot -2}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(c \cdot x\right)}}{s \cdot \left(c \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 67.6

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 67.6

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. pow-prod-down N/A

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

   16. lower-pow.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 79.6

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

   19. lift-pow.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 79.6

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

4. Applied rewrites 79.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

6. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

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

   4. lift-+.f64 97.7

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

8. Applied rewrites 97.7%

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

9. Final simplification 97.7%

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

Alternative 8: 79.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot c\_m\right) \cdot x\_m\\ \mathbf{if}\;{c\_m}^{2} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s\_m \cdot x\_m\right) \cdot \left(s\_m \cdot \left(c\_m \cdot x\_m\right)\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m c_m) x_m)))
   (if (<= (pow c_m 2.0) 5e-41)
     (/ 1.0 (* t_0 t_0))
     (/ 1.0 (* (* (* s_m x_m) (* s_m (* c_m x_m))) c_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * c_m) * x_m;
	double tmp;
	if (pow(c_m, 2.0) <= 5e-41) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((s_m * x_m) * (s_m * (c_m * x_m))) * c_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s_m * c_m) * x_m
    if ((c_m ** 2.0d0) <= 5d-41) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = 1.0d0 / (((s_m * x_m) * (s_m * (c_m * x_m))) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * c_m) * x_m;
	double tmp;
	if (Math.pow(c_m, 2.0) <= 5e-41) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((s_m * x_m) * (s_m * (c_m * x_m))) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (s_m * c_m) * x_m
	tmp = 0
	if math.pow(c_m, 2.0) <= 5e-41:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = 1.0 / (((s_m * x_m) * (s_m * (c_m * x_m))) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * c_m) * x_m)
	tmp = 0.0
	if ((c_m ^ 2.0) <= 5e-41)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s_m * x_m) * Float64(s_m * Float64(c_m * x_m))) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (s_m * c_m) * x_m;
	tmp = 0.0;
	if ((c_m ^ 2.0) <= 5e-41)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = 1.0 / (((s_m * x_m) * (s_m * (c_m * x_m))) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[N[Power[c$95$m, 2.0], $MachinePrecision], 5e-41], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 4.9999999999999996e-41

   1. Initial program 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 68.5

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 68.6

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l* N/A

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

      26. *-commutative N/A

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

      27. associate-*r* N/A

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

      28. lower-*.f64 N/A

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

      29. *-commutative N/A

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

      30. lower-*.f64 70.2

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

   9. Applied rewrites 70.2%

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

   if 4.9999999999999996e-41 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. associate-*l* N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*r* N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 N/A

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

   9. Applied rewrites 73.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lift-*.f64 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 N/A

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

       19. associate-*l* N/A

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

       20. lower-*.f64 N/A

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

       21. lower-*.f64 79.7

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

       22. lift-*.f64 N/A

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

       23. *-commutative N/A

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

       24. lower-*.f64 79.7

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

   11. Applied rewrites 79.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 64.2

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

5. Applied rewrites 64.2%

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

7. Applied rewrites 76.5%

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

8. Final simplification 76.5%

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

Alternative 10: 75.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 72.5

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. pow-prod-down N/A

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

   16. lower-pow.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 86.2

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

4. Applied rewrites 86.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 71.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 71.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 71.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 71.5

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

9. Applied rewrites 71.5%

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

Alternative 11: 77.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
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 97.4

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 76.3%

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

9. Final simplification 76.3%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))