mixedcos

Percentage Accurate: 66.5% → 96.9%

Time: 3.8s

Alternatives: 7

Speedup: 9.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 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 69.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.2

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

6. Applied rewrites 98.2%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]
7. 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. lower-+.f64 98.2

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

8. Applied rewrites 98.2%

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

Alternative 2: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot -2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)) (t_1 (* t_0 t_0)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -2e-194)
     (/ (* (* x x) -2.0) t_1)
     (/ 1.0 t_1))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double t_1 = t_0 * t_0;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-194) {
		tmp = ((x * x) * -2.0) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (c * s) * x
    t_1 = t_0 * t_0
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-2d-194)) then
        tmp = ((x * x) * (-2.0d0)) / t_1
    else
        tmp = 1.0d0 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double t_1 = t_0 * t_0;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -2e-194) {
		tmp = ((x * x) * -2.0) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	t_1 = t_0 * t_0
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -2e-194:
		tmp = ((x * x) * -2.0) / t_1
	else:
		tmp = 1.0 / t_1
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-194)
		tmp = Float64(Float64(Float64(x * x) * -2.0) / t_1);
	else
		tmp = Float64(1.0 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	t_1 = t_0 * t_0;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -2e-194)
		tmp = ((x * x) * -2.0) / t_1;
	else
		tmp = 1.0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $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], -2e-194], N[(N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 / t$95$1), $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))) < -2.00000000000000004e-194

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 37.6

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

   9. Applied rewrites 37.6%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 37.6

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

   12. Applied rewrites 37.6%

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

   if -2.00000000000000004e-194 < (/.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 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.5

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 63.1

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

   9. Applied rewrites 63.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 86.5%

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

Alternative 3: 71.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ t_1 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;x \leq 0.0205:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{t\_0}}{t\_0}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)) (t_1 (* (* c x) s)))
   (if (<= x 0.0205)
     (/ (/ (fma (* x x) -2.0 1.0) t_0) t_0)
     (if (<= x 2.25e+148)
       (/ (cos (+ x x)) (* (* (* c c) (* x x)) (* s s)))
       (/ 1.0 (* t_1 t_1))))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double t_1 = (c * x) * s;
	double tmp;
	if (x <= 0.0205) {
		tmp = (fma((x * x), -2.0, 1.0) / t_0) / t_0;
	} else if (x <= 2.25e+148) {
		tmp = cos((x + x)) / (((c * c) * (x * x)) * (s * s));
	} else {
		tmp = 1.0 / (t_1 * t_1);
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	t_1 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (x <= 0.0205)
		tmp = Float64(Float64(fma(Float64(x * x), -2.0, 1.0) / t_0) / t_0);
	elseif (x <= 2.25e+148)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(c * c) * Float64(x * x)) * Float64(s * s)));
	else
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[x, 0.0205], N[(N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 2.25e+148], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c * c), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0205000000000000009

   1. Initial program 67.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 75.0

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

   9. Applied rewrites 75.0%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 73.1

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

       6. cos-neg-rev 73.1

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

       7. distribute-lft-neg-in 73.1

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

       8. metadata-eval 73.1

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

       9. metadata-eval N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(c \cdot s\right)\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(c \cdot s\right) \cdot x\right)\right)}}{\left(c \cdot s\right) \cdot x} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite<=}\left(associate-*r*, \left(c \cdot \left(s \cdot x\right)\right)\right)}}{\left(c \cdot s\right) \cdot x} \]

       12. metadata-eval N/A

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

       13. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(s \cdot x\right) \cdot c\right)\right)}}{\left(c \cdot s\right) \cdot x} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(s \cdot x\right)\right) \cdot c}}{\left(c \cdot s\right) \cdot x} \]

       15. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(s \cdot x\right) \cdot c}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(c \cdot s\right)\right) \cdot x} \]

       16. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(s \cdot x\right) \cdot c}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(c \cdot s\right) \cdot x\right)\right)} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(s \cdot x\right) \cdot c}}{\mathsf{Rewrite<=}\left(associate-*r*, \left(c \cdot \left(s \cdot x\right)\right)\right)} \]

       18. metadata-eval N/A

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

       19. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(s \cdot x\right) \cdot c}}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(s \cdot x\right) \cdot c\right)\right)} \]

       20. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(s \cdot x\right) \cdot c}}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(s \cdot x\right)\right) \cdot c} \]

   11. Applied rewrites 73.1%

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

   if 0.0205000000000000009 < x < 2.24999999999999997e148

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.5

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

   6. Applied rewrites 76.5%

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

   if 2.24999999999999997e148 < x

   1. Initial program 68.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 47.9

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

      2. distribute-lft-neg-in 47.9

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

      3. cos-neg-rev 47.9

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

   7. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 59.7

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

   9. Applied rewrites 59.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. unswap-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 60.5

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

   11. Applied rewrites 60.5%

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

Alternative 4: 94.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return cos((x + x)) / (((c * s) * x) * (s * (c * 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((x + x)) / (((c * s) * x) * (s * (c * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.2

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

6. Applied rewrites 98.2%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]
7. 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. lower-+.f64 98.2

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

8. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 96.1

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

10. Applied rewrites 96.1%

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

Alternative 5: 78.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return 1.0 / (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 = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	return 1.0 / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 69.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.2

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in x around 0

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

   1. cos-neg-rev N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 60.2

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

9. Applied rewrites 60.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 76.8%

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

Alternative 6: 78.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	return 1.0 / (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 * x) * s
    code = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = (c * x) * s
	return 1.0 / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * x) * s;
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 69.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 58.9

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

4. Applied rewrites 58.9%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval 52.5

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

   2. distribute-lft-neg-in 52.5

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

   3. cos-neg-rev 52.5

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

7. Applied rewrites 52.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 67.1

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

9. Applied rewrites 67.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. unswap-sqr N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lift-*.f64 76.7

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

11. Applied rewrites 76.7%

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

Alternative 7: 68.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c x) (* (* c x) (* s s)))))

C

double code(double x, double c, double s) {
	return 1.0 / ((c * x) * ((c * x) * (s * 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 = 1.0d0 / ((c * x) * ((c * x) * (s * s)))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((c * x) * ((c * x) * (s * s)));
}

Python

def code(x, c, s):
	return 1.0 / ((c * x) * ((c * x) * (s * s)))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * x) * Float64(Float64(c * x) * Float64(s * s))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((c * x) * ((c * x) * (s * s)));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * x), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 58.9

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

4. Applied rewrites 58.9%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval 52.5

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

   2. distribute-lft-neg-in 52.5

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

   3. cos-neg-rev 52.5

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

7. Applied rewrites 52.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 67.1

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

9. Applied rewrites 67.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 68.2

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

11. Applied rewrites 68.2%

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

Reproduce

herbie shell --seed 2025064 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))