mixedcos

Percentage Accurate: 66.5% → 96.9%

Time: 3.3s

Alternatives: 8

Speedup: 4.2×

• 31.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 66.5% 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: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	return cos((x + x)) / pow((s_m * (c * x)), 2.0);
}

Fortran

s_m =     private
NOTE: x, c, 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, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = cos((x + x)) / ((s_m * (c * x)) ** 2.0d0)
end function

Java

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

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	return math.cos((x + x)) / math.pow((s_m * (c * x)), 2.0)

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	return Float64(cos(Float64(x + x)) / (Float64(s_m * Float64(c * x)) ^ 2.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 66.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 73.5

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 73.5

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

   14. lift-pow.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 73.5

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

3. Applied rewrites 73.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 82.5

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

5. Applied rewrites 82.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 73.2

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 73.2

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

7. Applied rewrites 73.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 96.9

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

9. Applied rewrites 96.9%

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

Alternative 2: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 3.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+115}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(c \cdot s\_m\right) \cdot \left(c \cdot s\_m\right)\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 3.8e-237)
     (/ 1.0 (* (* (* (* s_m (* s_m x)) x) c) c))
     (if (<= x 8e+115)
       (/ t_0 (* (* (* (* c s_m) (* c s_m)) x) x))
       (/ (/ t_0 s_m) (* (* (* c x) (* c x)) s_m))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 3.8e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 8e+115) {
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	} else {
		tmp = (t_0 / s_m) / (((c * x) * (c * x)) * s_m);
	}
	return tmp;
}

Fortran

s_m =     private
NOTE: x, c, 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, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (x <= 3.8d-237) then
        tmp = 1.0d0 / ((((s_m * (s_m * x)) * x) * c) * c)
    else if (x <= 8d+115) then
        tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x)
    else
        tmp = (t_0 / s_m) / (((c * x) * (c * x)) * s_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 3.8e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 8e+115) {
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	} else {
		tmp = (t_0 / s_m) / (((c * x) * (c * x)) * s_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 3.8e-237:
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c)
	elif x <= 8e+115:
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x)
	else:
		tmp = (t_0 / s_m) / (((c * x) * (c * x)) * s_m)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 3.8e-237)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x)) * x) * c) * c));
	elseif (x <= 8e+115)
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(c * s_m) * Float64(c * s_m)) * x) * x));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(Float64(Float64(c * x) * Float64(c * x)) * s_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 3.8e-237)
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	elseif (x <= 8e+115)
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	else
		tmp = (t_0 / s_m) / (((c * x) * (c * x)) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.8e-237], N[(1.0 / N[(N[(N[(N[(s$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+115], N[(t$95$0 / N[(N[(N[(N[(c * s$95$m), $MachinePrecision] * N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(N[(N[(c * x), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.80000000000000024e-237

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev 71.6

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

   10. Applied rewrites 71.6%

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

   if 3.80000000000000024e-237 < x < 8.0000000000000001e115

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if 8.0000000000000001e115 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 73.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 85.9

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

   9. Applied rewrites 85.9%

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

Alternative 3: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 3.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+115}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(c \cdot s\_m\right) \cdot \left(c \cdot s\_m\right)\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(s\_m \cdot s\_m\right) \cdot \left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 3.8e-237)
     (/ 1.0 (* (* (* (* s_m (* s_m x)) x) c) c))
     (if (<= x 8e+115)
       (/ t_0 (* (* (* (* c s_m) (* c s_m)) x) x))
       (/ t_0 (* (* s_m s_m) (* (* c x) (* c x))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 3.8e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 8e+115) {
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	} else {
		tmp = t_0 / ((s_m * s_m) * ((c * x) * (c * x)));
	}
	return tmp;
}

Fortran

s_m =     private
NOTE: x, c, 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, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (x <= 3.8d-237) then
        tmp = 1.0d0 / ((((s_m * (s_m * x)) * x) * c) * c)
    else if (x <= 8d+115) then
        tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x)
    else
        tmp = t_0 / ((s_m * s_m) * ((c * x) * (c * x)))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 3.8e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 8e+115) {
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	} else {
		tmp = t_0 / ((s_m * s_m) * ((c * x) * (c * x)));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 3.8e-237:
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c)
	elif x <= 8e+115:
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x)
	else:
		tmp = t_0 / ((s_m * s_m) * ((c * x) * (c * x)))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 3.8e-237)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x)) * x) * c) * c));
	elseif (x <= 8e+115)
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(c * s_m) * Float64(c * s_m)) * x) * x));
	else
		tmp = Float64(t_0 / Float64(Float64(s_m * s_m) * Float64(Float64(c * x) * Float64(c * x))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 3.8e-237)
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	elseif (x <= 8e+115)
		tmp = t_0 / ((((c * s_m) * (c * s_m)) * x) * x);
	else
		tmp = t_0 / ((s_m * s_m) * ((c * x) * (c * x)));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.8e-237], N[(1.0 / N[(N[(N[(N[(s$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+115], N[(t$95$0 / N[(N[(N[(N[(c * s$95$m), $MachinePrecision] * N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.80000000000000024e-237

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev 71.6

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

   10. Applied rewrites 71.6%

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

   if 3.80000000000000024e-237 < x < 8.0000000000000001e115

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if 8.0000000000000001e115 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 73.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.8

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 77.4

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

   9. Applied rewrites 77.4%

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

Alternative 4: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 3.52 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(c \cdot s\_m\right) \cdot \left(c \cdot s\_m\right)}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s\_m \cdot s\_m\right) \cdot \left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 4e-237)
   (/ 1.0 (* (* (* (* s_m (* s_m x)) x) c) c))
   (if (<= x 3.52e+17)
     (/ (/ (/ (fma -2.0 (* x x) 1.0) (* (* c s_m) (* c s_m))) x) x)
     (/ (cos (+ x x)) (* (* s_m s_m) (* (* c x) (* c x)))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 4e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 3.52e+17) {
		tmp = ((fma(-2.0, (x * x), 1.0) / ((c * s_m) * (c * s_m))) / x) / x;
	} else {
		tmp = cos((x + x)) / ((s_m * s_m) * ((c * x) * (c * x)));
	}
	return tmp;
}

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 4e-237)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x)) * x) * c) * c));
	elseif (x <= 3.52e+17)
		tmp = Float64(Float64(Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(c * s_m) * Float64(c * s_m))) / x) / x);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(s_m * s_m) * Float64(Float64(c * x) * Float64(c * x))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 4e-237], N[(1.0 / N[(N[(N[(N[(s$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.52e+17], N[(N[(N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(c * s$95$m), $MachinePrecision] * N[(c * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4e-237

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev 71.6

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

   10. Applied rewrites 71.6%

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

   if 4e-237 < x < 3.52e17

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 68.3

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 68.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.9

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

   6. Applied rewrites 53.9%

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

   if 3.52e17 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 73.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.8

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 77.4

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

   9. Applied rewrites 77.4%

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

Alternative 5: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 330:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(c \cdot s\_m\right) \cdot \left(c \cdot s\_m\right)}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(s\_m \cdot \left(\left(c \cdot \left(c \cdot x\right)\right) \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 4e-237)
   (/ 1.0 (* (* (* (* s_m (* s_m x)) x) c) c))
   (if (<= x 330.0)
     (/ (/ (/ (fma -2.0 (* x x) 1.0) (* (* c s_m) (* c s_m))) x) x)
     (/ (cos (+ x x)) (* s_m (* s_m (* (* c (* c x)) x)))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 4e-237) {
		tmp = 1.0 / ((((s_m * (s_m * x)) * x) * c) * c);
	} else if (x <= 330.0) {
		tmp = ((fma(-2.0, (x * x), 1.0) / ((c * s_m) * (c * s_m))) / x) / x;
	} else {
		tmp = cos((x + x)) / (s_m * (s_m * ((c * (c * x)) * x)));
	}
	return tmp;
}

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 4e-237)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x)) * x) * c) * c));
	elseif (x <= 330.0)
		tmp = Float64(Float64(Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(c * s_m) * Float64(c * s_m))) / x) / x);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(s_m * Float64(Float64(c * Float64(c * x)) * x))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 4e-237], N[(1.0 / N[(N[(N[(N[(s$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 330.0], N[(N[(N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(c * s$95$m), $MachinePrecision] * N[(c * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(s$95$m * N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4e-237

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev 71.6

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

   10. Applied rewrites 71.6%

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

   if 4e-237 < x < 330

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 68.3

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 68.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.9

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

   6. Applied rewrites 53.9%

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

   if 330 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 73.2

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

Alternative 6: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(\left(\left(s\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* (* (* s_m (* s_m x)) x) c) c)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -1e-203)
     (/ (fma -2.0 (* x x) 1.0) t_0)
     (/ 1.0 t_0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (((s_m * (s_m * x)) * x) * c) * c;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e-203) {
		tmp = fma(-2.0, (x * x), 1.0) / t_0;
	} else {
		tmp = 1.0 / t_0;
	}
	return tmp;
}

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(Float64(Float64(s_m * Float64(s_m * x)) * x) * c) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e-203)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / t_0);
	else
		tmp = Float64(1.0 / t_0);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(s$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-203], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-203

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

   if -1e-203 < (/.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 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.5

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.5

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

   3. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around 0

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

      1. cos-sum-rev 71.6

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

   10. Applied rewrites 71.6%

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

Alternative 7: 71.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 66.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 73.5

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 73.5

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

   14. lift-pow.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 73.5

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

3. Applied rewrites 73.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 82.5

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

5. Applied rewrites 82.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sin.f64 82.5

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

7. Applied rewrites 82.5%

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

8. Taylor expanded in x around 0

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

   1. cos-sum-rev 71.6

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

10. Applied rewrites 71.6%

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

Alternative 8: 64.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 66.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{{c}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot c\right) \cdot c}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot c\right) \cdot c}} \]

   10. lower-*.f64 73.5

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot c\right)} \cdot c} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   13. lower-*.f64 73.5

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\color{blue}{{s}^{2}} \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

   15. unpow2 N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

   16. lower-*.f64 73.5

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

3. Applied rewrites 73.5%

   \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\right) \cdot c}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(\left(s \cdot s\right) \cdot x\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot x\right) \cdot c\right) \cdot c} \]

   5. lower-*.f64 82.5

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(s \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot x\right) \cdot c\right) \cdot c} \]

5. Applied rewrites 82.5%

   \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot x\right) \cdot c\right) \cdot c} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(x + x\right)}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   3. cos-sum N/A

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x - \sin x \cdot \sin x}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x - \sin x \cdot \sin x}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x} - \sin x \cdot \sin x}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x} \cdot \cos x - \sin x \cdot \sin x}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{\cos x \cdot \color{blue}{\cos x} - \sin x \cdot \sin x}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\cos x \cdot \cos x - \color{blue}{\sin x \cdot \sin x}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\cos x \cdot \cos x - \color{blue}{\sin x} \cdot \sin x}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

   10. lower-sin.f64 82.5

       \[\leadsto \frac{\cos x \cdot \cos x - \sin x \cdot \color{blue}{\sin x}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

7. Applied rewrites 82.5%

   \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x - \sin x \cdot \sin x}}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c} \]

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. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. pow2 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]

   4. pow2 N/A

      \[\leadsto \frac{1}{\left(\left(s \cdot s\right) \cdot {x}^{2}\right) \cdot \left(c \cdot c\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(s \cdot s\right) \cdot {x}^{2}\right) \cdot \left(c \cdot c\right)} \]

   6. pow2 N/A

      \[\leadsto \frac{1}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot \left(c \cdot c\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot \left(c \cdot c\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{c} \cdot c\right)} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot \color{blue}{c}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot c} \]

   11. lift-*.f64 59.8

       \[\leadsto \frac{1}{\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot \color{blue}{c}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot c} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot c} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{\left(\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

   16. lower-*.f64 64.0

       \[\leadsto \frac{1}{\left(\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\right) \cdot c} \]

10. Applied rewrites 64.0%

    \[\leadsto \color{blue}{\frac{1}{\left(\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\right) \cdot c}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))