mixedcos

Percentage Accurate: 66.7% → 97.4%

Time: 3.9s

Alternatives: 9

Speedup: 4.2×

• 32.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c, s)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c, s)
use fmin_fmax_functions
    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: 97.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c_m * x) * s_m
    code = ((cos(x) + sin(x)) / t_0) * ((cos(x) - sin(x)) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-2 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 97.1

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

7. Applied rewrites 97.1%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. difference-of-squares N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unpow2 N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

9. Applied rewrites 97.4%

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

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{{\cos x}^{4}}{1} - \frac{{\sin x}^{4}}{1}}{{\left(s\_m \cdot \left(c\_m \cdot x\right)\right)}^{2}} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/
  (- (/ (pow (cos x) 4.0) 1.0) (/ (pow (sin x) 4.0) 1.0))
  (pow (* s_m (* c_m x)) 2.0)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return ((pow(cos(x), 4.0) / 1.0) - (pow(sin(x), 4.0) / 1.0)) / pow((s_m * (c_m * x)), 2.0);
}

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (((cos(x) ** 4.0d0) / 1.0d0) - ((sin(x) ** 4.0d0) / 1.0d0)) / ((s_m * (c_m * x)) ** 2.0d0)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return ((Math.pow(Math.cos(x), 4.0) / 1.0) - (Math.pow(Math.sin(x), 4.0) / 1.0)) / Math.pow((s_m * (c_m * x)), 2.0);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return ((math.pow(math.cos(x), 4.0) / 1.0) - (math.pow(math.sin(x), 4.0) / 1.0)) / math.pow((s_m * (c_m * x)), 2.0)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(Float64((cos(x) ^ 4.0) / 1.0) - Float64((sin(x) ^ 4.0) / 1.0)) / (Float64(s_m * Float64(c_m * x)) ^ 2.0))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (((cos(x) ^ 4.0) / 1.0) - ((sin(x) ^ 4.0) / 1.0)) / ((s_m * (c_m * x)) ^ 2.0);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(N[(N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision] / 1.0), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-2 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 97.1

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

7. Applied rewrites 97.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. flip-- N/A

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

   9. cos-sin-sum N/A

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

   10. div-sub N/A

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

   11. lower--.f64 N/A

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

9. Applied rewrites 97.0%

   \[\leadsto \frac{\color{blue}{\frac{{\cos x}^{4}}{1} - \frac{{\sin x}^{4}}{1}}}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}} \]
10. Add Preprocessing

Alternative 3: 97.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c_m * x) * s_m
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f64 97.1

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 97.1

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

7. Applied rewrites 97.1%

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

Alternative 4: 83.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;x \leq 3.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(s\_m \cdot s\_m\right) \cdot c\_m\right) \cdot c\_m\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(s\_m \cdot \left(c\_m \cdot x\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x) c_m)))
   (if (<= x 3.6e-8)
     (/ 1.0 (* t_0 t_0))
     (if (<= x 8.4e+130)
       (/ (cos (+ x x)) (* (* (* (* s_m s_m) c_m) c_m) (* x x)))
       (/ 1.0 (pow (* s_m (* c_m x)) 2.0))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double tmp;
	if (x <= 3.6e-8) {
		tmp = 1.0 / (t_0 * t_0);
	} else if (x <= 8.4e+130) {
		tmp = cos((x + x)) / ((((s_m * s_m) * c_m) * c_m) * (x * x));
	} else {
		tmp = 1.0 / pow((s_m * (c_m * x)), 2.0);
	}
	return tmp;
}

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s_m * x) * c_m
    if (x <= 3.6d-8) then
        tmp = 1.0d0 / (t_0 * t_0)
    else if (x <= 8.4d+130) then
        tmp = cos((x + x)) / ((((s_m * s_m) * c_m) * c_m) * (x * x))
    else
        tmp = 1.0d0 / ((s_m * (c_m * x)) ** 2.0d0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double tmp;
	if (x <= 3.6e-8) {
		tmp = 1.0 / (t_0 * t_0);
	} else if (x <= 8.4e+130) {
		tmp = Math.cos((x + x)) / ((((s_m * s_m) * c_m) * c_m) * (x * x));
	} else {
		tmp = 1.0 / Math.pow((s_m * (c_m * x)), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = (s_m * x) * c_m
	tmp = 0
	if x <= 3.6e-8:
		tmp = 1.0 / (t_0 * t_0)
	elif x <= 8.4e+130:
		tmp = math.cos((x + x)) / ((((s_m * s_m) * c_m) * c_m) * (x * x))
	else:
		tmp = 1.0 / math.pow((s_m * (c_m * x)), 2.0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (x <= 3.6e-8)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	elseif (x <= 8.4e+130)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(Float64(s_m * s_m) * c_m) * c_m) * Float64(x * x)));
	else
		tmp = Float64(1.0 / (Float64(s_m * Float64(c_m * x)) ^ 2.0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = (s_m * x) * c_m;
	tmp = 0.0;
	if (x <= 3.6e-8)
		tmp = 1.0 / (t_0 * t_0);
	elseif (x <= 8.4e+130)
		tmp = cos((x + x)) / ((((s_m * s_m) * c_m) * c_m) * (x * x));
	else
		tmp = 1.0 / ((s_m * (c_m * x)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x, 3.6e-8], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.4e+130], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.59999999999999981e-8

   1. Initial program 66.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-2 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around 0

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

      1. 2-cos N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. pow-prod-down N/A

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

      14. pow2 N/A

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

      15. unswap-sqr N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 80.5%

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

   if 3.59999999999999981e-8 < x < 8.39999999999999962e130

   1. Initial program 66.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if 8.39999999999999962e130 < x

   1. Initial program 66.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.5%

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

Alternative 5: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\right) \cdot \left(s\_m \cdot s\_m\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x) c_m)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x)))
        -5e-262)
     (/ (fma (* x x) -2.0 1.0) (* (* (* (* c_m c_m) x) (* s_m s_m)) x))
     (/ 1.0 (* t_0 t_0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -5e-262) {
		tmp = fma((x * x), -2.0, 1.0) / ((((c_m * c_m) * x) * (s_m * s_m)) * x);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(s_m * x) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -5e-262)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(Float64(Float64(c_m * c_m) * x) * Float64(s_m * s_m)) * x));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-262], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999992e-262

   1. Initial program 66.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 47.2

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

   6. Applied rewrites 47.2%

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

   if -4.99999999999999992e-262 < (/.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 66.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-2 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around 0

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

      1. 2-cos N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. pow-prod-down N/A

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

      14. pow2 N/A

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

      15. unswap-sqr N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 80.5%

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

Alternative 6: 80.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (s_m * x) * c_m
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-2 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 97.1

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in x around 0

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

   1. 2-cos N/A

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

   2. associate-*r* N/A

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

   3. unpow-prod-down N/A

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

   4. pow-prod-down N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. pow-prod-down N/A

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

   14. pow2 N/A

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

   15. unswap-sqr N/A

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

   16. lower-*.f64 N/A

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

10. Applied rewrites 80.5%

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

Alternative 7: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot c\_m\right)}\right) \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(\left(s\_m \cdot s\_m\right) \cdot c\_m\right) \cdot c\_m}\right) \cdot 0.6666666666666666\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x)))
      INFINITY)
   (* (* x (/ x (* (* s_m s_m) (* c_m c_m)))) 0.6666666666666666)
   (* (* x (/ x (* (* (* s_m s_m) c_m) c_m))) 0.6666666666666666)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / (((s_m * s_m) * c_m) * c_m))) * 0.6666666666666666;
	}
	return tmp;
}

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / (((s_m * s_m) * c_m) * c_m))) * 0.6666666666666666;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= math.inf:
		tmp = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666
	else:
		tmp = (x * (x / (((s_m * s_m) * c_m) * c_m))) * 0.6666666666666666
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(s_m * s_m) * Float64(c_m * c_m)))) * 0.6666666666666666);
	else
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(Float64(s_m * s_m) * c_m) * c_m))) * 0.6666666666666666);
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= Inf)
		tmp = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
	else
		tmp = (x * (x / (((s_m * s_m) * c_m) * c_m))) * 0.6666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * N[(x / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision], N[(N[(x * N[(x / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 66.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{\color{blue}{{x}^{2}}} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot \left(s \cdot s\right)} \cdot \frac{2}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot {c}^{2}} \cdot \frac{2}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      6. associate-*l* N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. pow2 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 26.8

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 26.8%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-*.f64 45.1

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 45.1%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       4. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       5. associate-*l* N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       6. unpow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       8. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       10. unpow2 N/A

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       11. lower-*.f64 44.4

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

   11. Applied rewrites 44.4%

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{\color{blue}{{x}^{2}}} \]

   4. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot \left(s \cdot s\right)} \cdot \frac{2}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot {c}^{2}} \cdot \frac{2}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      6. associate-*l* N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. pow2 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 26.8

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 26.8%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-*.f64 45.1

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 45.1%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 44.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \left(x \cdot \frac{x}{\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot c\_m\right)}\right) \cdot 0.6666666666666666 \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (* x (/ x (* (* s_m s_m) (* c_m c_m)))) 0.6666666666666666))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
}

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666d0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(x * Float64(x / Float64(Float64(s_m * s_m) * Float64(c_m * c_m)))) * 0.6666666666666666)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (x * (x / ((s_m * s_m) * (c_m * c_m)))) * 0.6666666666666666;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(x * N[(x / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{\color{blue}{{x}^{2}}} \]

4. Applied rewrites 28.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   3. pow2 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot \left(s \cdot s\right)} \cdot \frac{2}{3} \]

   4. *-commutative N/A

      \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot {c}^{2}} \cdot \frac{2}{3} \]

   5. pow2 N/A

      \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

   6. associate-*l* N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   11. pow2 N/A

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   12. lift-*.f64 26.8

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

7. Applied rewrites 26.8%

   \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   3. associate-/l* N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   5. lower-*.f64 45.1

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

9. Applied rewrites 45.1%

   \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    4. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    5. associate-*l* N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    6. unpow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    7. lower-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    8. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    9. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    10. unpow2 N/A

        \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    11. lower-*.f64 44.4

        \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

11. Applied rewrites 44.4%

    \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]
12. Add Preprocessing

Alternative 9: 39.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \left(x \cdot \frac{x}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\right) \cdot 0.6666666666666666 \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (* x (/ x (* (* s_m c_m) (* s_m c_m)))) 0.6666666666666666))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (x * (x / ((s_m * c_m) * (s_m * c_m)))) * 0.6666666666666666;
}

Fortran

c_m =     private
s_m =     private
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (x * (x / ((s_m * c_m) * (s_m * c_m)))) * 0.6666666666666666d0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (x * (x / ((s_m * c_m) * (s_m * c_m)))) * 0.6666666666666666;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (x * (x / ((s_m * c_m) * (s_m * c_m)))) * 0.6666666666666666

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(x * Float64(x / Float64(Float64(s_m * c_m) * Float64(s_m * c_m)))) * 0.6666666666666666)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (x * (x / ((s_m * c_m) * (s_m * c_m)))) * 0.6666666666666666;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(x * N[(x / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{\color{blue}{{x}^{2}}} \]

4. Applied rewrites 28.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   3. pow2 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot \left(s \cdot s\right)} \cdot \frac{2}{3} \]

   4. *-commutative N/A

      \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot {c}^{2}} \cdot \frac{2}{3} \]

   5. pow2 N/A

      \[\leadsto \frac{{x}^{2}}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

   6. associate-*l* N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{{x}^{2}}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   11. pow2 N/A

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   12. lift-*.f64 26.8

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

7. Applied rewrites 26.8%

   \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   3. associate-/l* N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   5. lower-*.f64 45.1

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

9. Applied rewrites 45.1%

   \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    4. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    5. associate-*l* N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    6. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    7. unswap-sqr N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right) \cdot \frac{2}{3} \]

    8. lower-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right) \cdot \frac{2}{3} \]

    9. lower-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right) \cdot \frac{2}{3} \]

    10. lower-*.f64 39.2

        \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right) \cdot 0.6666666666666666 \]

11. Applied rewrites 39.2%

    \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right) \cdot 0.6666666666666666 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))