mixedcos

Percentage Accurate: 66.8% → 99.2%

Time: 7.9s

Alternatives: 5

Speedup: 9.0×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot x\_m\right) \cdot {\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right)}^{-2}\\ \end{array} \end{array} \]

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 (<= x_m 4e-43)
   (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))
   (* (cos (* 2.0 x_m)) (pow (* (* c_m x_m) s_m) -2.0))))

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 tmp;
	if (x_m <= 4e-43) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = cos((2.0 * x_m)) * pow(((c_m * x_m) * s_m), -2.0);
	}
	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) :: tmp
    if (x_m <= 4d-43) then
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    else
        tmp = cos((2.0d0 * x_m)) * (((c_m * x_m) * s_m) ** (-2.0d0))
    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 tmp;
	if (x_m <= 4e-43) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = Math.cos((2.0 * x_m)) * Math.pow(((c_m * x_m) * s_m), -2.0);
	}
	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):
	tmp = 0
	if x_m <= 4e-43:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	else:
		tmp = math.cos((2.0 * x_m)) * math.pow(((c_m * x_m) * s_m), -2.0)
	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)
	tmp = 0.0
	if (x_m <= 4e-43)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) * (Float64(Float64(c_m * x_m) * s_m) ^ -2.0));
	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)
	tmp = 0.0;
	if (x_m <= 4e-43)
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	else
		tmp = cos((2.0 * x_m)) * (((c_m * x_m) * s_m) ^ -2.0);
	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_] := If[LessEqual[x$95$m, 4e-43], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000031e-43

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   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 82.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   10. 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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. lower-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   13. Applied rewrites 80.0%

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

   if 4.00000000000000031e-43 < x

   1. Initial program 68.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

   6. Applied rewrites 99.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot {\left(\left(c \cdot x\right) \cdot s\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -5e-142)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_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 = (c_m * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -5e-142) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_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(c_m * x_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))) <= -5e-142)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $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], -5e-142], 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[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $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))) < -5.0000000000000002e-142

   1. Initial program 48.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(x \cdot c\right) \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)} \]
   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 29.1

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

   8. Applied rewrites 29.1%

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

   if -5.0000000000000002e-142 < (/.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 66.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   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 86.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   10. 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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. lower-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 77.1

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

   11. Applied rewrites 77.1%

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

   13. Applied rewrites 83.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)))
   (if (<= x_m 3e-10)
     (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))
     (/ (/ (cos (* 2.0 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 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 3e-10) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = (cos((2.0 * 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 = (c_m * x_m) * s_m
    if (x_m <= 3d-10) then
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    else
        tmp = (cos((2.0d0 * 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 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 3e-10) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = (Math.cos((2.0 * 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 = (c_m * x_m) * s_m
	tmp = 0
	if x_m <= 3e-10:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	else:
		tmp = (math.cos((2.0 * 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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 3e-10)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x_m)) / 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 = (c_m * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 3e-10)
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	else
		tmp = (cos((2.0 * 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 3e-10], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e-10

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   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 83.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   10. 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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. lower-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 72.5

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

   11. Applied rewrites 72.5%

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

   13. Applied rewrites 80.2%

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

   if 3e-10 < x

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 83.8%

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

   6. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow-1 N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 99.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)))
   (if (<= x_m 3e-10)
     (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) 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 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 3e-10) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * 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 = (c_m * x_m) * s_m
    if (x_m <= 3d-10) then
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * 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 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 3e-10) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * 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 = (c_m * x_m) * s_m
	tmp = 0
	if x_m <= 3e-10:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * 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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 3e-10)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * 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 = (c_m * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 3e-10)
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 3e-10], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $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 < 3e-10

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   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 83.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   10. 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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*r* N/A

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

       17. lower-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 72.5

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

   11. Applied rewrites 72.5%

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

   13. Applied rewrites 80.2%

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

   if 3e-10 < x

   1. Initial program 68.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 99.0

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

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]
   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 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 9.0× speedup

Math

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

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) c_m) (* s_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) {
	return 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_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) * c_m) * (s_m * x_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) * c_m) * (s_m * x_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) * c_m) * (s_m * x_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) * c_m) * Float64(s_m * x_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) * c_m) * (s_m * x_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] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 96.8

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

5. Applied rewrites 96.8%

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

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 78.9%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
10. 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. *-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. *-commutative N/A

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

    15. unpow2 N/A

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

    16. associate-*r* N/A

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

    17. lower-*.f64 N/A

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

    18. *-commutative N/A

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

    19. lower-*.f64 70.4

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

11. Applied rewrites 70.4%

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

13. Applied rewrites 76.5%

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

14. Final simplification 76.5%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))