mixedcos

Percentage Accurate: 66.8% → 97.2%

Time: 8.2s

Alternatives: 13

Speedup: 2.4×

• 32.3% 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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 2.3× speedup

Math

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

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(Float64(cos(Float64(x + x)) / 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[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 85.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 93.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 93.5

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 93.5

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

6. Applied rewrites 93.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 2: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{-1} - 2}{\left(\left(s \cdot c\right) \cdot c\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-155)
     (/ (- (pow (* x x) -1.0) 2.0) (* (* (* s c) c) s))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = (pow((x * x), -1.0) - 2.0) / (((s * c) * c) * s);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s) * x
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-155)) then
        tmp = (((x * x) ** (-1.0d0)) - 2.0d0) / (((s * c) * c) * s)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = (Math.pow((x * x), -1.0) - 2.0) / (((s * c) * c) * s);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-155:
		tmp = (math.pow((x * x), -1.0) - 2.0) / (((s * c) * c) * s)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = Float64(Float64((Float64(x * x) ^ -1.0) - 2.0) / Float64(Float64(Float64(s * c) * c) * s));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = (((x * x) ^ -1.0) - 2.0) / (((s * c) * c) * s);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-155], N[(N[(N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(s * c), $MachinePrecision] * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-155

   1. Initial program 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

   7. Applied rewrites 56.7%

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

   if -1.00000000000000001e-155 < (/.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 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 94.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 94.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 94.1

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

   6. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 88.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{-1} - 2}{\left(\left(s \cdot c\right) \cdot c\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-155)
     (/ -2.0 (* s (* (* c c) s)))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s) * x
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-155)) then
        tmp = (-2.0d0) / (s * ((c * c) * s))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-155:
		tmp = -2.0 / (s * ((c * c) * s))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = Float64(-2.0 / Float64(s * Float64(Float64(c * c) * s)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = -2.0 / (s * ((c * c) * s));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-155], N[(-2.0 / N[(s * N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-155

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.7%

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

   10. Applied rewrites 56.7%

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

   12. Applied rewrites 56.7%

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

   if -1.00000000000000001e-155 < (/.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 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 94.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 94.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 94.1

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

   6. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 88.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot x\right) \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-155)
   (/ -2.0 (* s (* (* c c) s)))
   (/ 1.0 (* (* s x) (* c (* (* c s) x))))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = 1.0 / ((s * x) * (c * ((c * s) * x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-155)) then
        tmp = (-2.0d0) / (s * ((c * c) * s))
    else
        tmp = 1.0d0 / ((s * x) * (c * ((c * s) * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / (s * ((c * c) * s));
	} else {
		tmp = 1.0 / ((s * x) * (c * ((c * s) * x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-155:
		tmp = -2.0 / (s * ((c * c) * s))
	else:
		tmp = 1.0 / ((s * x) * (c * ((c * s) * x)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = Float64(-2.0 / Float64(s * Float64(Float64(c * c) * s)));
	else
		tmp = Float64(1.0 / Float64(Float64(s * x) * Float64(c * Float64(Float64(c * s) * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = -2.0 / (s * ((c * c) * s));
	else
		tmp = 1.0 / ((s * x) * (c * ((c * s) * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := 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], -1e-155], N[(-2.0 / N[(s * N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s * x), $MachinePrecision] * N[(c * N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-155

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.7%

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

   10. Applied rewrites 56.7%

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

   12. Applied rewrites 56.7%

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

   if -1.00000000000000001e-155 < (/.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 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 78.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. swap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot x\right) \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{s \cdot s}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{s}}{\left(c \cdot c\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) 0.0)
   (/ (/ -2.0 (* s s)) (* c c))
   (/ (/ 2.0 s) (* (* c c) s))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / (s * s)) / (c * c);
	} else {
		tmp = (2.0 / s) / ((c * c) * s);
	}
	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) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= 0.0d0) then
        tmp = ((-2.0d0) / (s * s)) / (c * c)
    else
        tmp = (2.0d0 / s) / ((c * c) * s)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / (s * s)) / (c * c);
	} else {
		tmp = (2.0 / s) / ((c * c) * s);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= 0.0:
		tmp = (-2.0 / (s * s)) / (c * c)
	else:
		tmp = (2.0 / s) / ((c * c) * s)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := 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], 0.0], N[(N[(-2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / s), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -0.0

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.7%

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

   10. Applied rewrites 60.0%

       \[\leadsto \frac{\frac{-2}{s \cdot s}}{c \cdot \color{blue}{c}} \]

   if -0.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.8%

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

   10. Applied rewrites 3.9%

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

   12. Applied rewrites 46.5%

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

Alternative 6: 49.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{s \cdot s}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) 0.0)
   (/ (/ -2.0 (* s s)) (* c c))
   (/ 2.0 (* s (* (* c c) s)))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / (s * s)) / (c * c);
	} else {
		tmp = 2.0 / (s * ((c * c) * s));
	}
	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) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= 0.0d0) then
        tmp = ((-2.0d0) / (s * s)) / (c * c)
    else
        tmp = 2.0d0 / (s * ((c * c) * s))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / (s * s)) / (c * c);
	} else {
		tmp = 2.0 / (s * ((c * c) * s));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= 0.0:
		tmp = (-2.0 / (s * s)) / (c * c)
	else:
		tmp = 2.0 / (s * ((c * c) * s))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := 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], 0.0], N[(N[(-2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(s * N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -0.0

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.7%

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

   10. Applied rewrites 60.0%

       \[\leadsto \frac{\frac{-2}{s \cdot s}}{c \cdot \color{blue}{c}} \]

   if -0.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.8%

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

   10. Applied rewrites 3.9%

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

   12. Applied rewrites 46.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{s \cdot s}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \left(\left(c \cdot c\right) \cdot s\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* (* c c) s))))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-155)
     (/ -2.0 t_0)
     (/ 2.0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = s * ((c * c) * s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / t_0;
	} else {
		tmp = 2.0 / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * ((c * c) * s)
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-155)) then
        tmp = (-2.0d0) / t_0
    else
        tmp = 2.0d0 / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = s * ((c * c) * s);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-155) {
		tmp = -2.0 / t_0;
	} else {
		tmp = 2.0 / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = s * ((c * c) * s)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-155:
		tmp = -2.0 / t_0
	else:
		tmp = 2.0 / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(s * Float64(Float64(c * c) * s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = Float64(-2.0 / t_0);
	else
		tmp = Float64(2.0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = s * ((c * c) * s);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-155)
		tmp = -2.0 / t_0;
	else
		tmp = 2.0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision]), $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], -1e-155], N[(-2.0 / t$95$0), $MachinePrecision], N[(2.0 / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-155

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.7%

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

   10. Applied rewrites 56.7%

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

   12. Applied rewrites 56.7%

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

   if -1.00000000000000001e-155 < (/.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 63.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.3%

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

   10. Applied rewrites 25.9%

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

   12. Applied rewrites 51.7%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;x \leq 0.116:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{t\_1}}{t\_1}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+141}:\\ \;\;\;\;\frac{t\_0}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot c\right) \cdot s\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (* (* c s) x)))
   (if (<= x 0.116)
     (/ (/ (fma (* x x) -2.0 1.0) t_1) t_1)
     (if (<= x 7e+141)
       (/ t_0 (* (* (* x x) c) (* (* s c) s)))
       (/ t_0 (* (* (* (* (* c x) x) c) s) s))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double t_1 = (c * s) * x;
	double tmp;
	if (x <= 0.116) {
		tmp = (fma((x * x), -2.0, 1.0) / t_1) / t_1;
	} else if (x <= 7e+141) {
		tmp = t_0 / (((x * x) * c) * ((s * c) * s));
	} else {
		tmp = t_0 / (((((c * x) * x) * c) * s) * s);
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (x <= 0.116)
		tmp = Float64(Float64(fma(Float64(x * x), -2.0, 1.0) / t_1) / t_1);
	elseif (x <= 7e+141)
		tmp = Float64(t_0 / Float64(Float64(Float64(x * x) * c) * Float64(Float64(s * c) * s)));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(Float64(c * x) * x) * c) * s) * s));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 0.116], N[(N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 7e+141], N[(t$95$0 / N[(N[(N[(x * x), $MachinePrecision] * c), $MachinePrecision] * N[(N[(s * c), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(N[(N[(c * x), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.116000000000000006

   1. Initial program 60.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 64.0%

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

   9. Applied rewrites 77.9%

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

   if 0.116000000000000006 < x < 6.9999999999999999e141

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 6.9999999999999999e141 < x

   1. Initial program 72.9%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 66.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 66.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 66.2

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

   6. Applied rewrites 66.2%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 84.0

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

   9. Applied rewrites 84.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.116:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+141}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot c\right) \cdot s\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;x \leq 0.116:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= x 0.116)
     (/ (/ (fma (* x x) -2.0 1.0) t_0) t_0)
     (/ (cos (* 2.0 x)) (* (* (* c x) x) (* (* c s) s))))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if (x <= 0.116) {
		tmp = (fma((x * x), -2.0, 1.0) / t_0) / t_0;
	} else {
		tmp = cos((2.0 * x)) / (((c * x) * x) * ((c * s) * s));
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (x <= 0.116)
		tmp = Float64(Float64(fma(Float64(x * x), -2.0, 1.0) / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64(c * x) * x) * Float64(Float64(c * s) * s)));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 0.116], N[(N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(c * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.116000000000000006

   1. Initial program 60.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 64.0%

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

   9. Applied rewrites 77.9%

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

   if 0.116000000000000006 < x

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 84.1

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

   4. Applied rewrites 84.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 84.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 84.1

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

   6. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 87.6

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 10: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot x\right) \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot c\right) \cdot s\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2e+106)
   (/ (cos (+ x x)) (* (* s x) (* c (* (* c s) x))))
   (/ (cos (* 2.0 x)) (* (* (* (* (* c x) x) c) s) s))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e+106) {
		tmp = cos((x + x)) / ((s * x) * (c * ((c * s) * x)));
	} else {
		tmp = cos((2.0 * x)) / (((((c * x) * x) * c) * s) * s);
	}
	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) :: tmp
    if (x <= 2d+106) then
        tmp = cos((x + x)) / ((s * x) * (c * ((c * s) * x)))
    else
        tmp = cos((2.0d0 * x)) / (((((c * x) * x) * c) * s) * s)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e+106) {
		tmp = Math.cos((x + x)) / ((s * x) * (c * ((c * s) * x)));
	} else {
		tmp = Math.cos((2.0 * x)) / (((((c * x) * x) * c) * s) * s);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2e+106:
		tmp = math.cos((x + x)) / ((s * x) * (c * ((c * s) * x)))
	else:
		tmp = math.cos((2.0 * x)) / (((((c * x) * x) * c) * s) * s)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2e+106)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(s * x) * Float64(c * Float64(Float64(c * s) * x))));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64(Float64(Float64(c * x) * x) * c) * s) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2e+106)
		tmp = cos((x + x)) / ((s * x) * (c * ((c * s) * x)));
	else
		tmp = cos((2.0 * x)) / (((((c * x) * x) * c) * s) * s);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2e+106], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * N[(c * N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(c * x), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000018e106

   1. Initial program 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 77.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 77.0

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

   6. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. swap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.7

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

   10. Applied rewrites 93.7%

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

   if 2.00000000000000018e106 < x

   1. Initial program 73.4%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 73.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 73.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 73.7

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

   6. Applied rewrites 73.7%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 79.8

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

   9. Applied rewrites 79.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot x\right) \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot c\right) \cdot s\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 76.5

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

4. Applied rewrites 76.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 76.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 76.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 76.5

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

6. Applied rewrites 76.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 81.3

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 95.1

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 95.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 95.1

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.1

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

8. Applied rewrites 95.1%

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

Alternative 12: 91.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 76.5

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

4. Applied rewrites 76.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 76.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 76.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 76.5

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

6. Applied rewrites 76.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 93.2

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 93.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 93.2

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

8. Applied rewrites 93.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 93.2

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

10. Applied rewrites 93.2%

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

Alternative 13: 27.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 27.6%

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

10. Applied rewrites 29.1%

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

12. Applied rewrites 29.2%

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

13. Final simplification 29.2%

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

Reproduce

herbie shell --seed 2024358 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))