mixedcos

Percentage Accurate: 66.7% → 97.6%

Time: 4.7s

Alternatives: 16

Speedup: 9.0×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% 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: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{{\cos x\_m}^{2} - {\sin x\_m}^{2}}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(-2 \cdot x\_m\right)}{{\left(c\_m \cdot x\_m\right)}^{2}}}{s\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.05e+29)
     (/ (/ (- (pow (cos x_m) 2.0) (pow (sin x_m) 2.0)) t_0) t_0)
     (/ (/ (/ (cos (* -2.0 x_m)) (pow (* c_m x_m) 2.0)) s_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.05e+29) {
		tmp = ((pow(cos(x_m), 2.0) - pow(sin(x_m), 2.0)) / t_0) / t_0;
	} else {
		tmp = ((cos((-2.0 * x_m)) / pow((c_m * x_m), 2.0)) / s_m) / s_m;
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    if (x_m <= 1.05d+29) then
        tmp = (((cos(x_m) ** 2.0d0) - (sin(x_m) ** 2.0d0)) / t_0) / t_0
    else
        tmp = ((cos(((-2.0d0) * x_m)) / ((c_m * x_m) ** 2.0d0)) / s_m) / s_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.05e+29) {
		tmp = ((Math.pow(Math.cos(x_m), 2.0) - Math.pow(Math.sin(x_m), 2.0)) / t_0) / t_0;
	} else {
		tmp = ((Math.cos((-2.0 * x_m)) / Math.pow((c_m * x_m), 2.0)) / s_m) / s_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1.05e+29:
		tmp = ((math.pow(math.cos(x_m), 2.0) - math.pow(math.sin(x_m), 2.0)) / t_0) / t_0
	else:
		tmp = ((math.cos((-2.0 * x_m)) / math.pow((c_m * x_m), 2.0)) / s_m) / s_m
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.05e+29)
		tmp = Float64(Float64(Float64((cos(x_m) ^ 2.0) - (sin(x_m) ^ 2.0)) / t_0) / t_0);
	else
		tmp = Float64(Float64(Float64(cos(Float64(-2.0 * x_m)) / (Float64(c_m * x_m) ^ 2.0)) / s_m) / s_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1.05e+29)
		tmp = (((cos(x_m) ^ 2.0) - (sin(x_m) ^ 2.0)) / t_0) / t_0;
	else
		tmp = ((cos((-2.0 * x_m)) / ((c_m * x_m) ^ 2.0)) / s_m) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05e+29], N[(N[(N[(N[Power[N[Cos[x$95$m], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Sin[x$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0500000000000001e29

   1. Initial program 69.0%

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

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. metadata-eval N/A

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

      6. cos-2 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.0500000000000001e29 < x

   1. Initial program 64.0%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 93.9

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

   3. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-prod-down N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

Alternative 2: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left(t\_0 \cdot x\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -2e-70)
     (/ (fma (* x_m x_m) -2.0 1.0) (* (* t_0 x_m) (* s_m c_m)))
     (pow t_0 -2.0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-70) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((t_0 * x_m) * (s_m * c_m));
	} else {
		tmp = pow(t_0, -2.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-70)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64(t_0 * x_m) * Float64(s_m * c_m)));
	else
		tmp = t_0 ^ -2.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-70], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(t$95$0 * x$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[t$95$0, -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))) < -1.99999999999999999e-70

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.8

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

   8. Applied rewrites 58.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-/.f64 N/A

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

      13. cos-neg-rev N/A

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

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

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

      15. metadata-eval N/A

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

      16. count-2-rev N/A

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

   10. Applied rewrites 58.8%

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

   if -1.99999999999999999e-70 < (/.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 65.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 75.1

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

   4. Applied rewrites 75.1%

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

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-prod-down N/A

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

   6. Applied rewrites 84.9%

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

Alternative 3: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left(t\_0 \cdot x\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -2e-70)
     (/ (fma (* x_m x_m) -2.0 1.0) (* (* t_0 x_m) (* s_m c_m)))
     (/ (/ 1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-70) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((t_0 * x_m) * (s_m * c_m));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-70)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64(t_0 * x_m) * Float64(s_m * c_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-70], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(t$95$0 * x$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $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))) < -1.99999999999999999e-70

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.8

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

   8. Applied rewrites 58.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-/.f64 N/A

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

      13. cos-neg-rev N/A

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

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

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

      15. metadata-eval N/A

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

      16. count-2-rev N/A

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

   10. Applied rewrites 58.8%

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

   if -1.99999999999999999e-70 < (/.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 65.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 75.1

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

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{{c}^{-2}}{{\left(s \cdot x\right)}^{2}}} \]
   5. 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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

Alternative 4: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -2e-70)
     (/ (fma (* x_m x_m) -2.0 1.0) (* (* (* c_m c_m) (* x_m x_m)) (* s_m s_m)))
     (/ (/ 1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-70) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-70)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64(Float64(c_m * c_m) * Float64(x_m * x_m)) * Float64(s_m * s_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-70], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $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))) < -1.99999999999999999e-70

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      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 62.8

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]
   5. 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 44.0

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

   6. Applied rewrites 44.0%

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

   if -1.99999999999999999e-70 < (/.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 65.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 75.1

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

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{{c}^{-2}}{{\left(s \cdot x\right)}^{2}}} \]
   5. 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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

Alternative 5: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(-2 \cdot x\_m\right)\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{{\left(c\_m \cdot x\_m\right)}^{2}}}{s\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* -2.0 x_m))) (t_1 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.05e+29)
     (/ (/ t_0 t_1) t_1)
     (/ (/ (/ t_0 (pow (* c_m x_m) 2.0)) s_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((-2.0 * x_m));
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.05e+29) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / pow((c_m * x_m), 2.0)) / s_m) / s_m;
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(((-2.0d0) * x_m))
    t_1 = (s_m * x_m) * c_m
    if (x_m <= 1.05d+29) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = ((t_0 / ((c_m * x_m) ** 2.0d0)) / s_m) / s_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((-2.0 * x_m));
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.05e+29) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / Math.pow((c_m * x_m), 2.0)) / s_m) / s_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((-2.0 * x_m))
	t_1 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1.05e+29:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = ((t_0 / math.pow((c_m * x_m), 2.0)) / s_m) / s_m
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(-2.0 * x_m))
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.05e+29)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(Float64(t_0 / (Float64(c_m * x_m) ^ 2.0)) / s_m) / s_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((-2.0 * x_m));
	t_1 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1.05e+29)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = ((t_0 / ((c_m * x_m) ^ 2.0)) / s_m) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05e+29], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 / N[Power[N[(c$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision] / s$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0500000000000001e29

   1. Initial program 69.0%

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

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 1.0500000000000001e29 < x

   1. Initial program 64.0%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 93.9

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

   3. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-prod-down N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

Alternative 6: 99.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(-s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 5.6 \cdot 10^{-78}:\\ \;\;\;\;{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(-2 \cdot x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (- s_m) (* c_m x_m))))
   (if (<= x_m 5.6e-78)
     (pow (* (* s_m x_m) c_m) -2.0)
     (/ (/ (cos (* -2.0 x_m)) t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = -s_m * (c_m * x_m);
	double tmp;
	if (x_m <= 5.6e-78) {
		tmp = pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = (cos((-2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -s_m * (c_m * x_m)
    if (x_m <= 5.6d-78) then
        tmp = ((s_m * x_m) * c_m) ** (-2.0d0)
    else
        tmp = (cos(((-2.0d0) * x_m)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = -s_m * (c_m * x_m);
	double tmp;
	if (x_m <= 5.6e-78) {
		tmp = Math.pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = (Math.cos((-2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = -s_m * (c_m * x_m)
	tmp = 0
	if x_m <= 5.6e-78:
		tmp = math.pow(((s_m * x_m) * c_m), -2.0)
	else:
		tmp = (math.cos((-2.0 * x_m)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(-s_m) * Float64(c_m * x_m))
	tmp = 0.0
	if (x_m <= 5.6e-78)
		tmp = Float64(Float64(s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(-2.0 * x_m)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = -s_m * (c_m * x_m);
	tmp = 0.0;
	if (x_m <= 5.6e-78)
		tmp = ((s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = (cos((-2.0 * x_m)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[((-s$95$m) * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.6e-78], N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.60000000000000047e-78

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 94.2

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

   4. Applied rewrites 94.2%

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

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-prod-down N/A

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

   6. Applied rewrites 99.7%

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

   if 5.60000000000000047e-78 < x

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 95.3

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

   3. Applied rewrites 95.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unswap-sqr N/A

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

      12. sqr-neg-rev N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. cos-neg-rev N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 99.2

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-neg.f64 99.2

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

      18. lift-neg.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 99.2%

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

Alternative 7: 98.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ t_1 := \left(c\_m \cdot x\_m\right) \cdot \left(-s\_m\right)\\ \mathbf{if}\;s\_m \leq 2.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(-2 \cdot x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)) (t_1 (* (* c_m x_m) (- s_m))))
   (if (<= s_m 2.5e+194)
     (/ (cos (* 2.0 x_m)) (* t_1 t_1))
     (/ (/ (cos (* -2.0 x_m)) t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double t_1 = (c_m * x_m) * -s_m;
	double tmp;
	if (s_m <= 2.5e+194) {
		tmp = cos((2.0 * x_m)) / (t_1 * t_1);
	} else {
		tmp = (cos((-2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    t_1 = (c_m * x_m) * -s_m
    if (s_m <= 2.5d+194) then
        tmp = cos((2.0d0 * x_m)) / (t_1 * t_1)
    else
        tmp = (cos(((-2.0d0) * x_m)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double t_1 = (c_m * x_m) * -s_m;
	double tmp;
	if (s_m <= 2.5e+194) {
		tmp = Math.cos((2.0 * x_m)) / (t_1 * t_1);
	} else {
		tmp = (Math.cos((-2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (s_m * x_m) * c_m
	t_1 = (c_m * x_m) * -s_m
	tmp = 0
	if s_m <= 2.5e+194:
		tmp = math.cos((2.0 * x_m)) / (t_1 * t_1)
	else:
		tmp = (math.cos((-2.0 * x_m)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	t_1 = Float64(Float64(c_m * x_m) * Float64(-s_m))
	tmp = 0.0
	if (s_m <= 2.5e+194)
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(cos(Float64(-2.0 * x_m)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (s_m * x_m) * c_m;
	t_1 = (c_m * x_m) * -s_m;
	tmp = 0.0;
	if (s_m <= 2.5e+194)
		tmp = cos((2.0 * x_m)) / (t_1 * t_1);
	else
		tmp = (cos((-2.0 * x_m)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * x$95$m), $MachinePrecision] * (-s$95$m)), $MachinePrecision]}, If[LessEqual[s$95$m, 2.5e+194], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < 2.49999999999999994e194

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 97.4

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

   3. Applied rewrites 97.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unswap-sqr N/A

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

      12. sqr-neg-rev N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 2.49999999999999994e194 < s

   1. Initial program 53.7%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 95.8

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

   3. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 8: 87.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ t_1 := \left(c\_m \cdot x\_m\right) \cdot \left(-s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x\_m \cdot x\_m, 0.6666666666666666\right), x\_m \cdot x\_m, -2\right), x\_m \cdot x\_m, 1\right)}{t\_0}}{t\_0}\\ \mathbf{elif}\;x\_m \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)) (t_1 (* (* c_m x_m) (- s_m))))
   (if (<= x_m 1.45)
     (/
      (/
       (fma
        (fma
         (fma -0.08888888888888889 (* x_m x_m) 0.6666666666666666)
         (* x_m x_m)
         -2.0)
        (* x_m x_m)
        1.0)
       t_0)
      t_0)
     (if (<= x_m 2.2e+140)
       (/ (cos (+ x_m x_m)) (* (* (* c_m (* s_m s_m)) (* x_m x_m)) c_m))
       (/ 1.0 (* t_1 t_1))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double t_1 = (c_m * x_m) * -s_m;
	double tmp;
	if (x_m <= 1.45) {
		tmp = (fma(fma(fma(-0.08888888888888889, (x_m * x_m), 0.6666666666666666), (x_m * x_m), -2.0), (x_m * x_m), 1.0) / t_0) / t_0;
	} else if (x_m <= 2.2e+140) {
		tmp = cos((x_m + x_m)) / (((c_m * (s_m * s_m)) * (x_m * x_m)) * c_m);
	} else {
		tmp = 1.0 / (t_1 * t_1);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	t_1 = Float64(Float64(c_m * x_m) * Float64(-s_m))
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(Float64(fma(fma(fma(-0.08888888888888889, Float64(x_m * x_m), 0.6666666666666666), Float64(x_m * x_m), -2.0), Float64(x_m * x_m), 1.0) / t_0) / t_0);
	elseif (x_m <= 2.2e+140)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(c_m * Float64(s_m * s_m)) * Float64(x_m * x_m)) * c_m));
	else
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * x$95$m), $MachinePrecision] * (-s$95$m)), $MachinePrecision]}, If[LessEqual[x$95$m, 1.45], N[(N[(N[(N[(N[(-0.08888888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$95$m, 2.2e+140], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.44999999999999996

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.4

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right)}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) + \color{blue}{1}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) \cdot {x}^{2} + 1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2, \color{blue}{{x}^{2}}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2 \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, {\color{blue}{x}}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + -2 \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + -2, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}, {x}^{2}, -2\right), {\color{blue}{x}}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      14. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      16. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), x \cdot \color{blue}{x}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      17. lift-*.f64 99.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot \color{blue}{x}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

   if 1.44999999999999996 < x < 2.1999999999999998e140

   1. Initial program 69.6%

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

      1. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow-prod-down N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-*.f64 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 77.3

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

   7. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.4

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

   9. Applied rewrites 84.4%

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

   if 2.1999999999999998e140 < x

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      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 52.9

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

   3. Applied rewrites 52.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]
   7. 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. sqr-neg-rev N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-neg.f64 68.7

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

   8. Applied rewrites 68.7%

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

Alternative 9: 85.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x\_m \cdot x\_m, 0.6666666666666666\right), x\_m \cdot x\_m, -2\right), x\_m \cdot x\_m, 1\right)}{t\_0}}{t\_0}\\ \mathbf{elif}\;x\_m \leq 1.35 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.6)
     (/
      (/
       (fma
        (fma
         (fma -0.08888888888888889 (* x_m x_m) 0.6666666666666666)
         (* x_m x_m)
         -2.0)
        (* x_m x_m)
        1.0)
       t_0)
      t_0)
     (if (<= x_m 1.35e+153)
       (/ (cos (+ x_m x_m)) (* (* (* c_m c_m) (* x_m x_m)) (* s_m s_m)))
       (/ 1.0 (* (* (* s_m s_m) (* c_m x_m)) (* c_m x_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.6) {
		tmp = (fma(fma(fma(-0.08888888888888889, (x_m * x_m), 0.6666666666666666), (x_m * x_m), -2.0), (x_m * x_m), 1.0) / t_0) / t_0;
	} else if (x_m <= 1.35e+153) {
		tmp = cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m));
	} else {
		tmp = 1.0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.6)
		tmp = Float64(Float64(fma(fma(fma(-0.08888888888888889, Float64(x_m * x_m), 0.6666666666666666), Float64(x_m * x_m), -2.0), Float64(x_m * x_m), 1.0) / t_0) / t_0);
	elseif (x_m <= 1.35e+153)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(c_m * c_m) * Float64(x_m * x_m)) * Float64(s_m * s_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x_m)) * Float64(c_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.6], N[(N[(N[(N[(N[(-0.08888888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$95$m, 1.35e+153], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6000000000000001

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.4

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. cos-neg-rev N/A

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

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

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right)}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) + \color{blue}{1}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) \cdot {x}^{2} + 1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2, \color{blue}{{x}^{2}}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2 \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, {\color{blue}{x}}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + -2 \cdot 1, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + -2, {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}, {x}^{2}, -2\right), {\color{blue}{x}}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      14. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), {x}^{2}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      16. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), x \cdot \color{blue}{x}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

      17. lift-*.f64 99.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot \color{blue}{x}, 1\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]

   if 1.6000000000000001 < x < 1.35e153

   1. Initial program 69.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. 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 73.9

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

   3. Applied rewrites 73.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}} \]
   4. 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 73.9

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

   5. Applied rewrites 73.9%

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

   if 1.35e153 < x

   1. Initial program 60.5%

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

      1. lift-*.f64 N/A

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

      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 51.2

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

   3. Applied rewrites 51.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]
   7. 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. unswap-sqr N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

Alternative 10: 99.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot \left(-s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{-52}:\\ \;\;\;\;{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* c_m x_m) (- s_m))))
   (if (<= x_m 2.3e-52)
     (pow (* (* s_m x_m) c_m) -2.0)
     (/ (cos (* 2.0 x_m)) (* t_0 t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (c_m * x_m) * -s_m;
	double tmp;
	if (x_m <= 2.3e-52) {
		tmp = pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = cos((2.0 * x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c_m * x_m) * -s_m
    if (x_m <= 2.3d-52) then
        tmp = ((s_m * x_m) * c_m) ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x_m)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (c_m * x_m) * -s_m;
	double tmp;
	if (x_m <= 2.3e-52) {
		tmp = Math.pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = Math.cos((2.0 * x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (c_m * x_m) * -s_m
	tmp = 0
	if x_m <= 2.3e-52:
		tmp = math.pow(((s_m * x_m) * c_m), -2.0)
	else:
		tmp = math.cos((2.0 * x_m)) / (t_0 * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(c_m * x_m) * Float64(-s_m))
	tmp = 0.0
	if (x_m <= 2.3e-52)
		tmp = Float64(Float64(s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (c_m * x_m) * -s_m;
	tmp = 0.0;
	if (x_m <= 2.3e-52)
		tmp = ((s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = cos((2.0 * x_m)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * x$95$m), $MachinePrecision] * (-s$95$m)), $MachinePrecision]}, If[LessEqual[x$95$m, 2.3e-52], N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999994e-52

   1. Initial program 66.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 92.9

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

   4. Applied rewrites 92.9%

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

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-prod-down N/A

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

   6. Applied rewrites 99.6%

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

   if 2.29999999999999994e-52 < x

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 95.0

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

   3. Applied rewrites 95.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unswap-sqr N/A

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

      12. sqr-neg-rev N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 11: 96.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.95 \cdot 10^{-32}:\\ \;\;\;\;{t\_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(t\_0 \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.95e-32)
     (pow t_0 -2.0)
     (/ (cos (* 2.0 x_m)) (* (* t_0 s_m) (* c_m x_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.95e-32) {
		tmp = pow(t_0, -2.0);
	} else {
		tmp = cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    if (x_m <= 1.95d-32) then
        tmp = t_0 ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x_m)) / ((t_0 * s_m) * (c_m * x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.95e-32) {
		tmp = Math.pow(t_0, -2.0);
	} else {
		tmp = Math.cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1.95e-32:
		tmp = math.pow(t_0, -2.0)
	else:
		tmp = math.cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.95e-32)
		tmp = t_0 ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(t_0 * s_m) * Float64(c_m * x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1.95e-32)
		tmp = t_0 ^ -2.0;
	else
		tmp = cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.95e-32], N[Power[t$95$0, -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$0 * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e-32

   1. Initial program 67.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   3. 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 92.0

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

   4. Applied rewrites 92.0%

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

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow-prod-down N/A

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

   6. Applied rewrites 99.6%

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

   if 1.9500000000000001e-32 < x

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      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. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 94.8

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

   3. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 12: 94.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(s\_m \cdot x\_m\right) \cdot \left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 2e+120)
   (/ (cos (+ x_m x_m)) (* (* (* s_m x_m) (* (* s_m x_m) c_m)) c_m))
   (/ (cos (* 2.0 x_m)) (* (* (* (* c_m x_m) c_m) x_m) (* s_m s_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2e+120) {
		tmp = cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m);
	} else {
		tmp = cos((2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * (s_m * s_m));
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2d+120) then
        tmp = cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m)
    else
        tmp = cos((2.0d0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * (s_m * s_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2e+120) {
		tmp = Math.cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m);
	} else {
		tmp = Math.cos((2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * (s_m * s_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2e+120:
		tmp = math.cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m)
	else:
		tmp = math.cos((2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * (s_m * s_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2e+120)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(s_m * x_m) * Float64(Float64(s_m * x_m) * c_m)) * c_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64(Float64(c_m * x_m) * c_m) * x_m) * Float64(s_m * s_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2e+120)
		tmp = cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m);
	else
		tmp = cos((2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * (s_m * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2e+120], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e120

   1. Initial program 68.9%

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

      1. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow-prod-down N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-*.f64 89.7

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

   3. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 89.7

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

   7. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 95.9

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

   9. Applied rewrites 95.9%

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

   if 2e120 < x

   1. Initial program 61.6%

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

      1. lift-*.f64 N/A

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

      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 55.2

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

   3. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

Alternative 13: 93.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = cos((x_m + x_m)) / (((s_m * x_m) * ((s_m * x_m) * c_m)) * c_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. lift-*.f64 N/A

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

   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. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. pow-prod-down N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-*.f64 85.9

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

3. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 85.9

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

5. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 85.9

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

7. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 93.7

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

9. Applied rewrites 93.7%

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

Alternative 14: 79.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m))) (/ (/ 1.0 t_0) t_0)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	return (1.0 / t_0) / t_0;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (s_m * x_m) * c_m
    code = (1.0d0 / t_0) / t_0
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	return (1.0 / t_0) / t_0;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (s_m * x_m) * c_m
	return (1.0 / t_0) / t_0

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = (s_m * x_m) * c_m;
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.7%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
3. 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 70.1

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

4. Applied rewrites 70.1%

   \[\leadsto \color{blue}{\frac{{c}^{-2}}{{\left(s \cdot x\right)}^{2}}} \]
5. 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. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 79.4

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

6. Applied rewrites 79.4%

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

Alternative 15: 79.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m)))) (/ 1.0 (* t_0 t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	return 1.0 / (t_0 * t_0);
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	return 1.0 / (t_0 * t_0);
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	return 1.0 / (t_0 * t_0)

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.7%

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

   1. lift-*.f64 N/A

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

   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 61.3

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

3. Applied rewrites 61.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 54.3%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]
7. 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. unpow2 N/A

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

   8. pow2 N/A

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

   9. pow-prod-down N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 79.3

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 79.3

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 79.3

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

8. Applied rewrites 79.3%

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

Alternative 16: 76.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * (s_m * (((s_m * x_m) * c_m) * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. lift-*.f64 N/A

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

   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 61.3

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

3. Applied rewrites 61.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 54.3%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]
7. 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. unswap-sqr N/A

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

   6. lift-*.f64 N/A

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

   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

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 76.1

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 76.1

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

8. Applied rewrites 76.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 76.7

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

10. Applied rewrites 76.7%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))