mixedcos

Percentage Accurate: 67.3% → 98.2%

Time: 6.4s

Alternatives: 15

Speedup: 2.4×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\_m\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{t\_0}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left({\left(c\_m \cdot x\_m\right)}^{2} \cdot s\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x_m))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 4.8e+89)
     (/ t_0 (* t_1 t_1))
     (/ t_0 (* (* (pow (* c_m x_m) 2.0) s_m) s_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((2.0 * x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4.8e+89) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / ((pow((c_m * x_m), 2.0) * s_m) * s_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x_m))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 4.8d+89) then
        tmp = t_0 / (t_1 * t_1)
    else
        tmp = t_0 / ((((c_m * x_m) ** 2.0d0) * s_m) * s_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4.8e+89) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / ((Math.pow((c_m * x_m), 2.0) * s_m) * s_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((2.0 * x_m))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 4.8e+89:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = t_0 / ((math.pow((c_m * x_m), 2.0) * s_m) * s_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(2.0 * x_m))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 4.8e+89)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(t_0 / Float64(Float64((Float64(c_m * x_m) ^ 2.0) * s_m) * s_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((2.0 * x_m));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 4.8e+89)
		tmp = t_0 / (t_1 * t_1);
	else
		tmp = t_0 / ((((c_m * x_m) ^ 2.0) * s_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4.8e+89], N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[Power[N[(c$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.80000000000000009e89

   1. Initial program 70.8%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.8

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 77.8

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

   4. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.8

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

   8. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* 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. lift-*.f64 N/A

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

      9. lower-*.f64 96.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 4.80000000000000009e89 < x

   1. Initial program 69.1%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 71.2

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 76.4

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. pow-prod-down N/A

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

      21. lower-pow.f64 N/A

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

      22. lower-*.f64 90.6

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

   6. Applied rewrites 90.6%

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

Alternative 2: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot x\_m, x\_m, 1\right)}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-127)
   (/ (fma (* -2.0 x_m) x_m 1.0) (* (* (* (* s_m (* s_m x_m)) x_m) c_m) c_m))
   (/ 1.0 (* (* (pow (* s_m x_m) 2.0) c_m) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = fma((-2.0 * x_m), x_m, 1.0) / ((((s_m * (s_m * x_m)) * x_m) * c_m) * c_m);
	} else {
		tmp = 1.0 / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = Float64(fma(Float64(-2.0 * x_m), x_m, 1.0) / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x_m)) * x_m) * c_m) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-127], N[(N[(N[(-2.0 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / N[(N[(N[(N[(s$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $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))) < -2.0000000000000001e-127

   1. Initial program 59.2%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 64.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 64.0

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 40.0

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

   9. Applied rewrites 40.0%

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

   if -2.0000000000000001e-127 < (/.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 71.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 85.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot x, x, 1\right)}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left({\left(s \cdot x\right)}^{2} \cdot c\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot x\_m, x\_m, 1\right)}{\left(\left(\left(s\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot x\_m\right) \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-127)
   (/ (fma (* -2.0 x_m) x_m 1.0) (* (* (* (* s_m (* s_m x_m)) x_m) c_m) c_m))
   (/ 1.0 (* (* (* (* (* s_m x_m) c_m) x_m) s_m) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = fma((-2.0 * x_m), x_m, 1.0) / ((((s_m * (s_m * x_m)) * x_m) * c_m) * c_m);
	} else {
		tmp = 1.0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = Float64(fma(Float64(-2.0 * x_m), x_m, 1.0) / Float64(Float64(Float64(Float64(s_m * Float64(s_m * x_m)) * x_m) * c_m) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * x_m) * s_m) * c_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-127], N[(N[(N[(-2.0 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / N[(N[(N[(N[(s$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $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))) < -2.0000000000000001e-127

   1. Initial program 59.2%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 64.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 64.0

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 40.0

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

   9. Applied rewrites 40.0%

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

   if -2.0000000000000001e-127 < (/.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 71.4%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 79.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.8

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.5%

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

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-2}{\left(c\_m \cdot s\_m\right) \cdot \left(c\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-127)
   (/ -2.0 (* (* c_m s_m) (* c_m s_m)))
   (/ 1.0 (* (* (* (* (* s_m x_m) c_m) x_m) s_m) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_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 ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-2d-127)) then
        tmp = (-2.0d0) / ((c_m * s_m) * (c_m * s_m))
    else
        tmp = 1.0d0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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 ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -2e-127:
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m))
	else:
		tmp = 1.0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = Float64(-2.0 / Float64(Float64(c_m * s_m) * Float64(c_m * s_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * x_m) * s_m) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
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 ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	else
		tmp = 1.0 / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-127], N[(-2.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $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))) < -2.0000000000000001e-127

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.8%

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

   10. Applied rewrites 39.8%

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

   12. Applied rewrites 40.1%

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

   if -2.0000000000000001e-127 < (/.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 71.4%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 79.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.8

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.5%

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

Alternative 5: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-2}{\left(c\_m \cdot s\_m\right) \cdot \left(c\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s\_m \cdot \left(\left(s\_m \cdot x\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-127)
   (/ -2.0 (* (* c_m s_m) (* c_m s_m)))
   (/ 1.0 (* (* s_m (* (* s_m x_m) (* c_m x_m))) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_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 ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-2d-127)) then
        tmp = (-2.0d0) / ((c_m * s_m) * (c_m * s_m))
    else
        tmp = 1.0d0 / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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 ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -2e-127:
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m))
	else:
		tmp = 1.0 / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = Float64(-2.0 / Float64(Float64(c_m * s_m) * Float64(c_m * s_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(s_m * Float64(Float64(s_m * x_m) * Float64(c_m * x_m))) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
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 ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	else
		tmp = 1.0 / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-127], N[(-2.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s$95$m * N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$m), $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))) < -2.0000000000000001e-127

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.8%

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

   10. Applied rewrites 39.8%

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

   12. Applied rewrites 40.1%

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

   if -2.0000000000000001e-127 < (/.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 71.4%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 79.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.5

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.4%

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

Alternative 6: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-2}{\left(c\_m \cdot s\_m\right) \cdot \left(c\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot s\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-127)
   (/ -2.0 (* (* c_m s_m) (* c_m s_m)))
   (/ 1.0 (* (* (* (* (* c_m c_m) x_m) x_m) s_m) s_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_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 ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-2d-127)) then
        tmp = (-2.0d0) / ((c_m * s_m) * (c_m * s_m))
    else
        tmp = 1.0d0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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 ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -2e-127) {
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	} else {
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -2e-127:
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m))
	else:
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = Float64(-2.0 / Float64(Float64(c_m * s_m) * Float64(c_m * s_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c_m * c_m) * x_m) * x_m) * s_m) * s_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
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 ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -2e-127)
		tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
	else
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-127], N[(-2.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $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))) < -2.0000000000000001e-127

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.8%

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

   10. Applied rewrites 39.8%

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

   12. Applied rewrites 40.1%

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

   if -2.0000000000000001e-127 < (/.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 71.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.7%

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

Alternative 7: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 7.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(-2 \cdot x\_m\right)}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot s\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 7.2e+140)
     (/ (cos (* 2.0 x_m)) (* t_0 t_0))
     (/ (/ (cos (* -2.0 x_m)) (* (* (* (* c_m x_m) c_m) x_m) s_m)) s_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
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 <= 7.2e+140) {
		tmp = cos((2.0 * x_m)) / (t_0 * t_0);
	} else {
		tmp = (cos((-2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * s_m)) / s_m;
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_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 <= 7.2d+140) then
        tmp = cos((2.0d0 * x_m)) / (t_0 * t_0)
    else
        tmp = (cos(((-2.0d0) * x_m)) / ((((c_m * x_m) * c_m) * x_m) * s_m)) / s_m
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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 <= 7.2e+140) {
		tmp = Math.cos((2.0 * x_m)) / (t_0 * t_0);
	} else {
		tmp = (Math.cos((-2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * s_m)) / s_m;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[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 <= 7.2e+140:
		tmp = math.cos((2.0 * x_m)) / (t_0 * t_0)
	else:
		tmp = (math.cos((-2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * s_m)) / s_m
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 7.2e+140)
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(cos(Float64(-2.0 * x_m)) / Float64(Float64(Float64(Float64(c_m * x_m) * c_m) * x_m) * s_m)) / s_m);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
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 <= 7.2e+140)
		tmp = cos((2.0 * x_m)) / (t_0 * t_0);
	else
		tmp = (cos((-2.0 * x_m)) / ((((c_m * x_m) * c_m) * x_m) * s_m)) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 7.2e+140], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(-2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.1999999999999999e140

   1. Initial program 71.0%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 78.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.2

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.1

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

   8. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* 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. lift-*.f64 N/A

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

      9. lower-*.f64 96.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.4

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.4

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

   10. Applied rewrites 96.4%

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

   if 7.1999999999999999e140 < x

   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. Taylor expanded in x around inf

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.7%

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

   10. Applied rewrites 88.4%

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

Alternative 8: 95.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right) \cdot \left(x\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (<= c_m 2e-190)
   (/ (cos (+ x_m x_m)) (* (* (* (* (* s_m x_m) c_m) x_m) s_m) c_m))
   (/ (cos (* 2.0 x_m)) (* (* c_m (* c_m (* x_m s_m))) (* x_m s_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 2e-190) {
		tmp = cos((x_m + x_m)) / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	} else {
		tmp = cos((2.0 * x_m)) / ((c_m * (c_m * (x_m * s_m))) * (x_m * s_m));
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_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 (c_m <= 2d-190) then
        tmp = cos((x_m + x_m)) / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m)
    else
        tmp = cos((2.0d0 * x_m)) / ((c_m * (c_m * (x_m * s_m))) * (x_m * s_m))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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 (c_m <= 2e-190) {
		tmp = Math.cos((x_m + x_m)) / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	} else {
		tmp = Math.cos((2.0 * x_m)) / ((c_m * (c_m * (x_m * s_m))) * (x_m * s_m));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if c_m <= 2e-190:
		tmp = math.cos((x_m + x_m)) / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m)
	else:
		tmp = math.cos((2.0 * x_m)) / ((c_m * (c_m * (x_m * s_m))) * (x_m * s_m))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 (c_m <= 2e-190)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * x_m) * s_m) * c_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(c_m * Float64(c_m * Float64(x_m * s_m))) * Float64(x_m * s_m)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
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 (c_m <= 2e-190)
		tmp = cos((x_m + x_m)) / (((((s_m * x_m) * c_m) * x_m) * s_m) * c_m);
	else
		tmp = cos((2.0 * x_m)) / ((c_m * (c_m * (x_m * s_m))) * (x_m * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[c$95$m, 2e-190], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2e-190

   1. Initial program 68.1%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 78.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.1

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.9

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

   10. Applied rewrites 90.9%

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

   if 2e-190 < c

   1. Initial program 73.7%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 78.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.7

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.7

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

   4. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.8

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

   6. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.8

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.7

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

   8. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 97.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 97.2

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

   10. Applied rewrites 97.2%

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

Alternative 9: 95.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{s\_m \cdot \left(\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m\right) \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 3e-15)
   (/ (fma (* x_m x_m) -2.0 1.0) (pow (* c_m (* x_m s_m)) 2.0))
   (/ (cos (* 2.0 x_m)) (* s_m (* (* (* (* s_m x_m) c_m) x_m) c_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
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 <= 3e-15) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = cos((2.0 * x_m)) / (s_m * ((((s_m * x_m) * c_m) * x_m) * c_m));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 <= 3e-15)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(s_m * Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * x_m) * c_m)));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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, 3e-15], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e-15

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 72.2%

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

   if 3e-15 < x

   1. Initial program 71.0%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 81.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.3

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 81.3

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

   4. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 91.7

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 91.6

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

   8. Applied rewrites 91.6%

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

Alternative 10: 93.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(s\_m \cdot \left(\left(s\_m \cdot x\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 3.5e-51)
   (/ (fma (* x_m x_m) -2.0 1.0) (pow (* c_m (* x_m s_m)) 2.0))
   (/ (cos (+ x_m x_m)) (* (* s_m (* (* s_m x_m) (* c_m x_m))) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
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 <= 3.5e-51) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = cos((x_m + x_m)) / ((s_m * ((s_m * x_m) * (c_m * x_m))) * c_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
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 <= 3.5e-51)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(s_m * Float64(Float64(s_m * x_m) * Float64(c_m * x_m))) * c_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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, 3.5e-51], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999997e-51

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 68.3%

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

   9. Applied rewrites 71.0%

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

   if 3.4999999999999997e-51 < x

   1. Initial program 73.1%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 82.2

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.3

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

   6. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 91.3

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

   8. Applied rewrites 91.3%

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

Alternative 11: 97.1% accurate, 2.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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)))) (/ (cos (* 2.0 x_m)) (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
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);
	return cos((2.0 * x_m)) / (t_0 * t_0);
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x_m * s_m)
    code = cos((2.0d0 * x_m)) / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
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);
	return Math.cos((2.0 * x_m)) / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[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)
	return math.cos((2.0 * x_m)) / (t_0 * t_0)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	return Float64(cos(Float64(2.0 * x_m)) / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = cos((2.0 * x_m)) / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 70.5%

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

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 78.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 78.4

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 78.4

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

4. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 91.2

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

6. Applied rewrites 91.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 91.2

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 91.3

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

8. Applied rewrites 91.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* 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. lift-*.f64 N/A

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

   9. lower-*.f64 96.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 96.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 96.8

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 96.8

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

10. Applied rewrites 96.8%

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

Alternative 12: 31.1% accurate, 12.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (/ -2.0 (* (* (* s_m s_m) c_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((s_m * s_m) * c_m) * c_m);
}

Fortran

s_m =     private
c_m =     private
x_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 = (-2.0d0) / (((s_m * s_m) * c_m) * c_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((s_m * s_m) * c_m) * c_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (((s_m * s_m) * c_m) * c_m)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(Float64(Float64(s_m * s_m) * c_m) * c_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / (((s_m * s_m) * c_m) * c_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[(-2.0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.0%

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

10. Applied rewrites 29.0%

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

12. Applied rewrites 30.8%

    \[\leadsto \frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \]
13. Add Preprocessing

Alternative 13: 28.1% accurate, 12.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (/ -2.0 (* (* (* c_m s_m) s_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((c_m * s_m) * s_m) * c_m);
}

Fortran

s_m =     private
c_m =     private
x_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 = (-2.0d0) / (((c_m * s_m) * s_m) * c_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((c_m * s_m) * s_m) * c_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (((c_m * s_m) * s_m) * c_m)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(Float64(Float64(c_m * s_m) * s_m) * c_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / (((c_m * s_m) * s_m) * c_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[(-2.0 / N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.0%

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

10. Applied rewrites 29.0%

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

12. Applied rewrites 28.2%

    \[\leadsto \frac{-2}{\left(\left(c \cdot s\right) \cdot s\right) \cdot c} \]
13. Add Preprocessing

Alternative 14: 27.8% accurate, 12.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (/ -2.0 (* (* s_m s_m) (* c_m c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * s_m) * (c_m * c_m));
}

Fortran

s_m =     private
c_m =     private
x_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 = (-2.0d0) / ((s_m * s_m) * (c_m * c_m))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * s_m) * (c_m * c_m));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / ((s_m * s_m) * (c_m * c_m))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(Float64(s_m * s_m) * Float64(c_m * c_m)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / ((s_m * s_m) * (c_m * c_m));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[(-2.0 / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.0%

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

10. Applied rewrites 29.0%

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

Alternative 15: 25.4% accurate, 12.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
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 (/ -2.0 (* (* c_m s_m) (* c_m s_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((c_m * s_m) * (c_m * s_m));
}

Fortran

s_m =     private
c_m =     private
x_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 = (-2.0d0) / ((c_m * s_m) * (c_m * s_m))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((c_m * s_m) * (c_m * s_m));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / ((c_m * s_m) * (c_m * s_m))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(Float64(c_m * s_m) * Float64(c_m * s_m)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / ((c_m * s_m) * (c_m * s_m));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $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[(-2.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.0%

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

10. Applied rewrites 29.0%

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

12. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2025019 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))