mixedcos

Percentage Accurate: 66.6% → 99.3%

Time: 4.8s

Alternatives: 12

Speedup: 9.0×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-42}:\\ \;\;\;\;{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 5e-42)
     (pow (* (* s_m x_m) c_m) -2.0)
     (/ (cos (+ x_m 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 <= 5e-42) {
		tmp = pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = cos((x_m + 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 <= 5d-42) then
        tmp = ((s_m * x_m) * c_m) ** (-2.0d0)
    else
        tmp = cos((x_m + 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 <= 5e-42) {
		tmp = Math.pow(((s_m * x_m) * c_m), -2.0);
	} else {
		tmp = Math.cos((x_m + 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 <= 5e-42:
		tmp = math.pow(((s_m * x_m) * c_m), -2.0)
	else:
		tmp = math.cos((x_m + 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) * s_m)
	tmp = 0.0
	if (x_m <= 5e-42)
		tmp = Float64(Float64(s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m + 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 <= 5e-42)
		tmp = ((s_m * x_m) * c_m) ^ -2.0;
	else
		tmp = cos((x_m + 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, 5e-42], N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000003e-42

   1. Initial program 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. pow-prod-down N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-unpow N/A

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

      9. pow-neg N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

   7. Applied rewrites 85.2%

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

   if 5.00000000000000003e-42 < x

   1. Initial program 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-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 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-pow.f64 N/A

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

      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 97.4

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

   6. Applied rewrites 97.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-42}:\\ \;\;\;\;{\left(\left(s \cdot x\right) \cdot c\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{2}}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -5e-119)
     (/ (fma (* x_m x_m) -2.0 1.0) (* t_0 t_0))
     (/ 1.0 (pow (* (* s_m x_m) c_m) 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 = (c_m * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-119) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / pow(((s_m * x_m) * c_m), 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(c_m * x_m) * s_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))) <= -5e-119)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / (Float64(Float64(s_m * x_m) * c_m) ^ 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[(c$95$m * x$95$m), $MachinePrecision] * s$95$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], -5e-119], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999993e-119

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      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 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 56.1

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

   9. Applied rewrites 56.1%

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

   if -4.99999999999999993e-119 < (/.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 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 85.9

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

      2. distribute-lft-neg-in 85.9

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

      3. cos-neg-rev 85.9

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

   7. Applied rewrites 85.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s \cdot x\right) \cdot c\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{-2}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -5e-119)
     (/ (fma (* x_m x_m) -2.0 1.0) (* t_0 t_0))
     (pow (* (* s_m x_m) c_m) -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 = (c_m * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-119) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / (t_0 * t_0);
	} else {
		tmp = pow(((s_m * x_m) * c_m), -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(c_m * x_m) * s_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))) <= -5e-119)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(s_m * x_m) * c_m) ^ -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[(c$95$m * x$95$m), $MachinePrecision] * s$95$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], -5e-119], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999993e-119

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      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 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 56.1

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

   9. Applied rewrites 56.1%

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

   if -4.99999999999999993e-119 < (/.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 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. pow-prod-down N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-unpow N/A

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

      9. pow-neg N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

   7. Applied rewrites 85.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(s \cdot x\right) \cdot c\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)) (t_1 (* (* c_m s_m) x_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -5e-119)
     (/ (fma (* x_m x_m) -2.0 1.0) (* t_0 t_0))
     (/ 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 = (c_m * x_m) * s_m;
	double t_1 = (c_m * s_m) * x_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-119) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / (t_0 * t_0);
	} 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(c_m * x_m) * s_m)
	t_1 = Float64(Float64(c_m * s_m) * x_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))) <= -5e-119)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(t_0 * t_0));
	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[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$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], -5e-119], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999993e-119

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      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 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 56.1

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

   9. Applied rewrites 56.1%

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

   if -4.99999999999999993e-119 < (/.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 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 61.4

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

      2. distribute-lft-neg-in 61.4

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

      3. cos-neg-rev 61.4

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

   7. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow-prod-down N/A

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

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. unpow2 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 84.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -5e-119)
     (/ (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 = (c_m * s_m) * x_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-119) {
		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(c_m * s_m) * x_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))) <= -5e-119)
		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(1.0 / Float64(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[(c$95$m * s$95$m), $MachinePrecision] * x$95$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], -5e-119], 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[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999993e-119

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. cos-neg-rev N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 42.2

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

   7. Applied rewrites 42.2%

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

   if -4.99999999999999993e-119 < (/.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 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 61.4

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

      2. distribute-lft-neg-in 61.4

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

      3. cos-neg-rev 61.4

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

   7. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow-prod-down N/A

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

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. unpow2 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 84.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{2}}\\ \mathbf{elif}\;x\_m \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot s\_m\right) \cdot \left(\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

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 (cos (* 2.0 x_m))))
   (if (<= x_m 5e-140)
     (/ 1.0 (pow (* (* s_m x_m) c_m) 2.0))
     (if (<= x_m 5.6e+149)
       (/ t_0 (* (* c_m s_m) (* (* c_m s_m) (* x_m x_m))))
       (/ t_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 = cos((2.0 * x_m));
	double tmp;
	if (x_m <= 5e-140) {
		tmp = 1.0 / pow(((s_m * x_m) * c_m), 2.0);
	} else if (x_m <= 5.6e+149) {
		tmp = t_0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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 = cos((2.0d0 * x_m))
    if (x_m <= 5d-140) then
        tmp = 1.0d0 / (((s_m * x_m) * c_m) ** 2.0d0)
    else if (x_m <= 5.6d+149) then
        tmp = t_0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)))
    else
        tmp = t_0 / (((s_m * s_m) * (c_m * x_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 = Math.cos((2.0 * x_m));
	double tmp;
	if (x_m <= 5e-140) {
		tmp = 1.0 / Math.pow(((s_m * x_m) * c_m), 2.0);
	} else if (x_m <= 5.6e+149) {
		tmp = t_0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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 = math.cos((2.0 * x_m))
	tmp = 0
	if x_m <= 5e-140:
		tmp = 1.0 / math.pow(((s_m * x_m) * c_m), 2.0)
	elif x_m <= 5.6e+149:
		tmp = t_0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)))
	else:
		tmp = t_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 = cos(Float64(2.0 * x_m))
	tmp = 0.0
	if (x_m <= 5e-140)
		tmp = Float64(1.0 / (Float64(Float64(s_m * x_m) * c_m) ^ 2.0));
	elseif (x_m <= 5.6e+149)
		tmp = Float64(t_0 / Float64(Float64(c_m * s_m) * Float64(Float64(c_m * s_m) * Float64(x_m * x_m))));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x_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 = cos((2.0 * x_m));
	tmp = 0.0;
	if (x_m <= 5e-140)
		tmp = 1.0 / (((s_m * x_m) * c_m) ^ 2.0);
	elseif (x_m <= 5.6e+149)
		tmp = t_0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	else
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 5e-140], N[(1.0 / N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 5.6e+149], N[(t$95$0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$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 < 5.00000000000000015e-140

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 83.8

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

      2. distribute-lft-neg-in 83.8

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

      3. cos-neg-rev 83.8

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

   7. Applied rewrites 83.8%

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

   if 5.00000000000000015e-140 < x < 5.5999999999999998e149

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. 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 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-pow.f64 N/A

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

      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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if 5.5999999999999998e149 < x

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. swap-sqr N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 75.4

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

   6. Applied rewrites 75.4%

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

Alternative 7: 95.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;s\_m \leq 2.3 \cdot 10^{+174}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(t\_0 \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{-2}\\ \end{array} \end{array} \]

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 (<= s_m 2.3e+174)
     (/ (cos (* 2.0 x_m)) (* (* t_0 s_m) (* c_m x_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 (s_m <= 2.3e+174) {
		tmp = cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	} else {
		tmp = pow(t_0, -2.0);
	}
	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 * x_m) * c_m
    if (s_m <= 2.3d+174) then
        tmp = cos((2.0d0 * x_m)) / ((t_0 * s_m) * (c_m * x_m))
    else
        tmp = t_0 ** (-2.0d0)
    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 (s_m <= 2.3e+174) {
		tmp = Math.cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	} else {
		tmp = Math.pow(t_0, -2.0);
	}
	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
	tmp = 0
	if s_m <= 2.3e+174:
		tmp = math.cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m))
	else:
		tmp = math.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 (s_m <= 2.3e+174)
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(t_0 * s_m) * Float64(c_m * x_m)));
	else
		tmp = t_0 ^ -2.0;
	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;
	tmp = 0.0;
	if (s_m <= 2.3e+174)
		tmp = cos((2.0 * x_m)) / ((t_0 * s_m) * (c_m * x_m));
	else
		tmp = t_0 ^ -2.0;
	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]}, If[LessEqual[s$95$m, 2.3e+174], 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], N[Power[t$95$0, -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 2.2999999999999998e174

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-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 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-pow.f64 N/A

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

      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 93.2

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

   6. Applied rewrites 93.2%

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

   if 2.2999999999999998e174 < s

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. 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 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. pow-prod-down N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-unpow N/A

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

      9. pow-neg N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

   7. Applied rewrites 95.1%

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

Alternative 8: 96.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \frac{\frac{\cos \left(-2 \cdot x\_m\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

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))) (/ (/ (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;
	return (cos((-2.0 * x_m)) / 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 = (cos(((-2.0d0) * x_m)) / 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 (Math.cos((-2.0 * x_m)) / 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 (math.cos((-2.0 * x_m)) / 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(cos(Float64(-2.0 * x_m)) / 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 = (cos((-2.0 * x_m)) / 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[(N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 67.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   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 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-/.f64 N/A

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

   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. metadata-eval N/A

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

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

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

   13. cos-neg-rev N/A

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

   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 97.8

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

6. Applied rewrites 97.8%

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

Alternative 9: 94.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(\left(c\_m \cdot x\_m\right) \cdot x\_m\right)}\\ \end{array} \end{array} \]

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
 (if (<= x_m 2.1e-46)
   (/ 1.0 (pow (* (* s_m x_m) c_m) 2.0))
   (/ (cos (+ x_m x_m)) (* (* s_m (* s_m c_m)) (* (* c_m x_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 tmp;
	if (x_m <= 2.1e-46) {
		tmp = 1.0 / pow(((s_m * x_m) * c_m), 2.0);
	} else {
		tmp = cos((x_m + x_m)) / ((s_m * (s_m * c_m)) * ((c_m * x_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) :: tmp
    if (x_m <= 2.1d-46) then
        tmp = 1.0d0 / (((s_m * x_m) * c_m) ** 2.0d0)
    else
        tmp = cos((x_m + x_m)) / ((s_m * (s_m * c_m)) * ((c_m * x_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 tmp;
	if (x_m <= 2.1e-46) {
		tmp = 1.0 / Math.pow(((s_m * x_m) * c_m), 2.0);
	} else {
		tmp = Math.cos((x_m + x_m)) / ((s_m * (s_m * c_m)) * ((c_m * x_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):
	tmp = 0
	if x_m <= 2.1e-46:
		tmp = 1.0 / math.pow(((s_m * x_m) * c_m), 2.0)
	else:
		tmp = math.cos((x_m + x_m)) / ((s_m * (s_m * c_m)) * ((c_m * x_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)
	tmp = 0.0
	if (x_m <= 2.1e-46)
		tmp = Float64(1.0 / (Float64(Float64(s_m * x_m) * c_m) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(s_m * Float64(s_m * c_m)) * Float64(Float64(c_m * x_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)
	tmp = 0.0;
	if (x_m <= 2.1e-46)
		tmp = 1.0 / (((s_m * x_m) * c_m) ^ 2.0);
	else
		tmp = cos((x_m + x_m)) / ((s_m * (s_m * c_m)) * ((c_m * x_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_] := If[LessEqual[x$95$m, 2.1e-46], N[(1.0 / N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(c$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999987e-46

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 85.1

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

      2. distribute-lft-neg-in 85.1

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

      3. cos-neg-rev 85.1

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

   7. Applied rewrites 85.1%

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

   if 2.09999999999999987e-46 < x

   1. Initial program 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-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 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-pow.f64 N/A

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

      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. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 80.0

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

   8. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.5

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

   10. Applied rewrites 84.5%

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

Alternative 10: 80.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;c\_m \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)) (t_1 (* (* c_m s_m) x_m)))
   (if (<= c_m 8e-58) (/ 1.0 (* t_1 t_1)) (/ (/ 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 * x_m) * s_m;
	double t_1 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 8e-58) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = (1.0 / 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 = (c_m * x_m) * s_m
    t_1 = (c_m * s_m) * x_m
    if (c_m <= 8d-58) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = (1.0d0 / 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 t_1 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 8e-58) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = (1.0 / 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
	t_1 = (c_m * s_m) * x_m
	tmp = 0
	if c_m <= 8e-58:
		tmp = 1.0 / (t_1 * t_1)
	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(c_m * x_m) * s_m)
	t_1 = Float64(Float64(c_m * s_m) * x_m)
	tmp = 0.0
	if (c_m <= 8e-58)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(1.0 / 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;
	t_1 = (c_m * s_m) * x_m;
	tmp = 0.0;
	if (c_m <= 8e-58)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = (1.0 / 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]}, Block[{t$95$1 = N[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 8e-58], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 8.0000000000000002e-58

   1. Initial program 64.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 55.3

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

      2. distribute-lft-neg-in 55.3

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

      3. cos-neg-rev 55.3

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

   7. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow-prod-down N/A

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

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. unpow2 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 77.2%

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

   if 8.0000000000000002e-58 < c

   1. Initial program 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 62.9

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

      2. distribute-lft-neg-in 62.9

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

      3. cos-neg-rev 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. pow-prod-down N/A

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

      14. unpow-prod-down N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. unpow2 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. associate-/r* N/A

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

   9. Applied rewrites 88.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;c\_m \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

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 (* (* c_m s_m) x_m)))
   (if (<= c_m 5.8e-58)
     (/ 1.0 (* t_0 t_0))
     (/ 1.0 (* (* (* (* c_m x_m) s_m) c_m) (* s_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 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 5.8e-58) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / ((((c_m * x_m) * s_m) * c_m) * (s_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 = (c_m * s_m) * x_m
    if (c_m <= 5.8d-58) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = 1.0d0 / ((((c_m * x_m) * s_m) * c_m) * (s_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 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 5.8e-58) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / ((((c_m * x_m) * s_m) * c_m) * (s_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 = (c_m * s_m) * x_m
	tmp = 0
	if c_m <= 5.8e-58:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = 1.0 / ((((c_m * x_m) * s_m) * c_m) * (s_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(c_m * s_m) * x_m)
	tmp = 0.0
	if (c_m <= 5.8e-58)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * c_m) * Float64(s_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 = (c_m * s_m) * x_m;
	tmp = 0.0;
	if (c_m <= 5.8e-58)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = 1.0 / ((((c_m * x_m) * s_m) * c_m) * (s_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[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 5.8e-58], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 5.7999999999999998e-58

   1. Initial program 64.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 55.3

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

      2. distribute-lft-neg-in 55.3

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

      3. cos-neg-rev 55.3

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

   7. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow-prod-down N/A

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

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. unpow2 N/A

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

      17. sqr-neg-rev N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 77.2%

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

   if 5.7999999999999998e-58 < c

   1. Initial program 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval 62.9

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

      2. distribute-lft-neg-in 62.9

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

      3. cos-neg-rev 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      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

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 87.0

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

   9. Applied rewrites 87.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot c\right) \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.6% 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 67.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 62.9

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

4. Applied rewrites 62.9%

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

5. Taylor expanded in x around 0

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

   1. metadata-eval 57.8

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

   2. distribute-lft-neg-in 57.8

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

   3. cos-neg-rev 57.8

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

7. Applied rewrites 57.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. swap-sqr N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 72.0

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

9. Applied rewrites 72.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. associate-*r* N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f64 N/A

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

    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. lower-*.f64 N/A

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

    14. lower-*.f64 78.3

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

11. Applied rewrites 78.3%

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

Reproduce

herbie shell --seed 2025084 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))