mixedcos

Percentage Accurate: 67.3% → 97.2%

Time: 4.6s

Alternatives: 11

Speedup: 9.0×

• 31.5% 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: 67.3% 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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (c * s) * x
    code = (cos(((-2.0d0) * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = (cos((-2.0 * 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[(-2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.0

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

4. Applied rewrites 97.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. lower-*.f64 97.8

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 97.8

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

6. Applied rewrites 97.8%

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

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x)))))
   (if (<= t_0 (- INFINITY))
     (/ (fma (* x x) -2.0 1.0) (* (* (* s x) (* s x)) (* c c)))
     (if (<= t_0 -2e-241)
       (/ (cos (+ x x)) (* (* (* c c) (* x x)) (* s s)))
       (pow (* (* c s) x) -2.0)))))

C

double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((x * x), -2.0, 1.0) / (((s * x) * (s * x)) * (c * c));
	} else if (t_0 <= -2e-241) {
		tmp = cos((x + x)) / (((c * c) * (x * x)) * (s * s));
	} else {
		tmp = pow(((c * s) * x), -2.0);
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(Float64(s * x) * Float64(s * x)) * Float64(c * c)));
	elseif (t_0 <= -2e-241)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(c * c) * Float64(x * x)) * Float64(s * s)));
	else
		tmp = Float64(Float64(c * s) * x) ^ -2.0;
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(s * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-241], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c * c), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision], -2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if -inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.9999999999999999e-241

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 79.0

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

   6. Applied rewrites 79.0%

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

   if -1.9999999999999999e-241 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. associate-/r* 75.2

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

      2. *-commutative 75.2

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

      3. pow2 75.2

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

      4. associate-*r* 75.2

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

      5. pow2 75.2

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

      6. *-commutative 75.2

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

      7. pow2 75.2

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

      8. pow-prod-down 75.2

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-pow.f64 75.2

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

   7. Applied rewrites 87.9%

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

Alternative 3: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -2e-241)
   (/ (fma (* x x) -2.0 1.0) (* (* (* s x) (* s x)) (* c c)))
   (pow (* (* c s) x) -2.0)))

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))) <= -2e-241) {
		tmp = fma((x * x), -2.0, 1.0) / (((s * x) * (s * x)) * (c * c));
	} else {
		tmp = pow(((c * s) * x), -2.0);
	}
	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))) <= -2e-241)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(Float64(s * x) * Float64(s * x)) * Float64(c * c)));
	else
		tmp = Float64(Float64(c * s) * x) ^ -2.0;
	end
	return 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], -2e-241], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(s * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision], -2.0], $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))) < -1.9999999999999999e-241

   1. Initial program 78.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 37.9

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

   9. Applied rewrites 37.9%

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

   if -1.9999999999999999e-241 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. associate-/r* 75.2

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

      2. *-commutative 75.2

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

      3. pow2 75.2

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

      4. associate-*r* 75.2

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

      5. pow2 75.2

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

      6. *-commutative 75.2

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

      7. pow2 75.2

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

      8. pow-prod-down 75.2

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-pow.f64 75.2

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

   7. Applied rewrites 87.9%

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

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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))) -2e-241)
     (/ (fma (* x x) -2.0 1.0) (* (* (* s x) (* s x)) (* c c)))
     (/ (/ 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))) <= -2e-241) {
		tmp = fma((x * x), -2.0, 1.0) / (((s * x) * (s * x)) * (c * c));
	} 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))) <= -2e-241)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(Float64(s * x) * Float64(s * x)) * Float64(c * c)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return 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], -2e-241], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(s * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(c * c), $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))) < -1.9999999999999999e-241

   1. Initial program 78.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 37.9

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

   9. Applied rewrites 37.9%

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

   if -1.9999999999999999e-241 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 87.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 87.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot -2}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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))) -2e-241)
     (/ (* (* x x) -2.0) (* (* (* c c) (* x x)) (* s 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))) <= -2e-241) {
		tmp = ((x * x) * -2.0) / (((c * c) * (x * x)) * (s * 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))) <= (-2d-241)) then
        tmp = ((x * x) * (-2.0d0)) / (((c * c) * (x * x)) * (s * 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))) <= -2e-241) {
		tmp = ((x * x) * -2.0) / (((c * c) * (x * x)) * (s * 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))) <= -2e-241:
		tmp = ((x * x) * -2.0) / (((c * c) * (x * x)) * (s * 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))) <= -2e-241)
		tmp = Float64(Float64(Float64(x * x) * -2.0) / Float64(Float64(Float64(c * c) * Float64(x * x)) * Float64(s * 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))) <= -2e-241)
		tmp = ((x * x) * -2.0) / (((c * c) * (x * x)) * (s * 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], -2e-241], N[(N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(c * c), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(s * 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))) < -1.9999999999999999e-241

   1. Initial program 78.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 52.0

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 19.7

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

   7. Applied rewrites 19.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.7

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

   10. Applied rewrites 19.7%

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

   if -1.9999999999999999e-241 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 87.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 87.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot -2}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.0

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

4. Applied rewrites 97.0%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 97.0

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.0

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.0

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

6. Applied rewrites 97.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 97.0

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

8. Applied rewrites 97.0%

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

Alternative 7: 76.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(\left(s \cdot x\right) \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot c\right) \cdot \left(s \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 5.4e-192)
   (/ 1.0 (* c (* s (* x (* (* s x) c)))))
   (/ 1.0 (* (* (* (* c x) s) c) (* s x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 5.4e-192) {
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	} else {
		tmp = 1.0 / ((((c * x) * s) * 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 (c <= 5.4d-192) then
        tmp = 1.0d0 / (c * (s * (x * ((s * x) * c))))
    else
        tmp = 1.0d0 / ((((c * x) * s) * c) * (s * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 5.4e-192) {
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	} else {
		tmp = 1.0 / ((((c * x) * s) * c) * (s * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 5.4e-192:
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))))
	else:
		tmp = 1.0 / ((((c * x) * s) * c) * (s * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 5.4e-192)
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(x * Float64(Float64(s * x) * c)))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * x) * s) * c) * Float64(s * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 5.4e-192)
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	else
		tmp = 1.0 / ((((c * x) * s) * c) * (s * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 5.4e-192], N[(1.0 / N[(c * N[(s * N[(x * N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision] * c), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.39999999999999982e-192

   1. Initial program 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

   9. Applied rewrites 77.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 78.4

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 79.0

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

   11. Applied rewrites 79.0%

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

   if 5.39999999999999982e-192 < c

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 58.1

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 N/A

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

      25. lift-*.f64 78.5

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

   9. Applied rewrites 78.5%

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

Alternative 8: 77.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;c \leq 3 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(\left(s \cdot x\right) \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s)))
   (if (<= c 3e-192)
     (/ 1.0 (* c (* s (* x (* (* s x) c)))))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if (c <= 3e-192) {
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	} 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 * x) * s
    if (c <= 3d-192) then
        tmp = 1.0d0 / (c * (s * (x * ((s * x) * c))))
    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 * x) * s;
	double tmp;
	if (c <= 3e-192) {
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * x) * s
	tmp = 0
	if c <= 3e-192:
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (c <= 3e-192)
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(x * Float64(Float64(s * x) * c)))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * x) * s;
	tmp = 0.0;
	if (c <= 3e-192)
		tmp = 1.0 / (c * (s * (x * ((s * x) * c))));
	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 * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[c, 3e-192], N[(1.0 / N[(c * N[(s * N[(x * N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.9999999999999999e-192

   1. Initial program 65.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

   9. Applied rewrites 77.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 78.2

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 78.8

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

   11. Applied rewrites 78.8%

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

   if 2.9999999999999999e-192 < c

   1. Initial program 70.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 57.5

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-prod-down N/A

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

      12. unpow-prod-down N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 77.7

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 77.7

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

      25. lift-*.f64 N/A

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

      26. lift-*.f64 N/A

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

      27. associate-*r* N/A

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

      28. *-commutative N/A

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

   9. Applied rewrites 80.6%

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

Alternative 9: 78.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f64 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 68.6

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

5. Applied rewrites 68.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow-prod-down N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-/r* N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 80.1

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 80.1

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 80.1

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

7. Applied rewrites 80.1%

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

8. Final simplification 80.1%

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

Alternative 10: 78.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 57.9

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

4. Applied rewrites 57.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 52.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. unpow-prod-down N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. sqr-neg-rev N/A

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

   18. lower-*.f64 N/A

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

9. Applied rewrites 80.1%

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

10. Final simplification 80.1%

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

Alternative 11: 75.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 57.9

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

4. Applied rewrites 57.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 52.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

   7. unswap-sqr N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. associate-*r* N/A

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

   23. *-commutative N/A

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

   24. associate-*r* N/A

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

   25. lower-*.f64 N/A

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

9. Applied rewrites 76.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 77.0

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. associate-*r* N/A

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

    13. *-commutative N/A

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

    14. lower-*.f64 N/A

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

    15. lift-*.f64 78.5

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

11. Applied rewrites 78.5%

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))