mixedcos

Percentage Accurate: 66.6% → 97.2%

Time: 6.6s

Alternatives: 6

Speedup: 2.4×

• 30.8% 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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot x\\ \frac{\frac{\cos \left(-2 \cdot x\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) x))) (/ (/ (cos (* -2.0 x)) t_0) t_0)))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	return (cos((-2.0 * x)) / t_0) / t_0;
}

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
    real(8) :: t_0
    t_0 = (s * c) * x
    code = (cos(((-2.0d0) * x)) / t_0) / t_0
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	return (Math.cos((-2.0 * x)) / t_0) / t_0;
}

Python

def code(x, c, s):
	t_0 = (s * c) * x
	return (math.cos((-2.0 * x)) / t_0) / t_0

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * c) * x)
	return Float64(Float64(cos(Float64(-2.0 * x)) / t_0) / t_0)
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (s * c) * x;
	tmp = (cos((-2.0 * x)) / t_0) / t_0;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[Cos[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 70.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 79.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 79.8

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 79.8

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

4. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 94.5

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 94.5

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

6. Applied rewrites 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 98.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.9

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 98.9

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 98.9

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

   25. lift-*.f64 N/A

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

   26. *-commutative N/A

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

   27. lower-*.f64 98.9

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

8. Applied rewrites 98.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.0

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-neg-in N/A

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

   10. cos-neg-rev N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 99.0

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 99.0

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 99.0

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

10. Applied rewrites 99.0%

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

Alternative 2: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-233}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -5e-233)
     (/ (- 2.0) (* s (* (* c c) s)))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -5e-233) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s) * x
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-5d-233)) then
        tmp = -2.0d0 / (s * ((c * c) * s))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -5e-233) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -5e-233:
		tmp = -2.0 / (s * ((c * c) * s))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -5e-233)
		tmp = Float64(Float64(-2.0) / Float64(s * Float64(Float64(c * c) * s)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -5e-233)
		tmp = -2.0 / (s * ((c * c) * s));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $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, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-233], N[((-2.0) / N[(s * N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -5.00000000000000012e-233

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.1%

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

   10. Applied rewrites 59.1%

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

   12. Applied rewrites 59.1%

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

   if -5.00000000000000012e-233 < (/.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 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.1

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 79.1

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

   4. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 93.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.9

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

   6. Applied rewrites 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.6

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 88.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-233}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= x 5.6e-46)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (+ x x)) (* (* x c) (* s (* s (* c x))))))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if (x <= 5.6e-46) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / ((x * c) * (s * (s * (c * x))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s) * x
    if (x <= 5.6d-46) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x + x)) / ((x * c) * (s * (s * (c * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if (x <= 5.6e-46) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x + x)) / ((x * c) * (s * (s * (c * x))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	tmp = 0
	if x <= 5.6e-46:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x + x)) / ((x * c) * (s * (s * (c * x))))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (x <= 5.6e-46)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(x * c) * Float64(s * Float64(s * Float64(c * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 0.0;
	if (x <= 5.6e-46)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x + x)) / ((x * c) * (s * (s * (c * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 5.6e-46], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * c), $MachinePrecision] * N[(s * N[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999997e-46

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.2

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 93.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 99.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.1

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.1

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 87.3%

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

   if 5.5999999999999997e-46 < x

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 81.1

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 81.1

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

   4. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.8

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

   8. Applied rewrites 96.8%

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

Alternative 4: 97.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x))) (/ (cos (+ x x)) (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return cos((x + x)) / (t_0 * t_0);
}

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
    real(8) :: t_0
    t_0 = (c * s) * x
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	return math.cos((x + x)) / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 79.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 79.8

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 79.8

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

4. Applied rewrites 79.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 94.5

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 94.5

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

6. Applied rewrites 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 98.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.9

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 98.9

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 98.9

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

   25. lift-*.f64 N/A

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

   26. *-commutative N/A

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

   27. lower-*.f64 98.9

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

8. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f64 98.9

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

10. Applied rewrites 98.9%

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

Alternative 5: 28.2% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

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 = ((-2.0d0) / ((s * s) * c)) / c
end function

Java

public static double code(double x, double c, double s) {
	return (-2.0 / ((s * s) * c)) / c;
}

Python

def code(x, c, s):
	return (-2.0 / ((s * s) * c)) / c

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 28.7%

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

10. Applied rewrites 32.0%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 31.4%

    \[\leadsto \frac{\frac{-2}{\left(s \cdot s\right) \cdot c}}{\color{blue}{c}} \]
14. Add Preprocessing

Alternative 6: 28.4% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

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 = -2.0d0 / (s * ((c * c) * s))
end function

Java

public static double code(double x, double c, double s) {
	return -2.0 / (s * ((c * c) * s));
}

Python

def code(x, c, s):
	return -2.0 / (s * ((c * c) * s))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 28.7%

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

10. Applied rewrites 32.0%

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

12. Applied rewrites 33.8%

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

13. Final simplification 33.8%

    \[\leadsto \frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))