mixedcos

Percentage Accurate: 66.1% → 98.4%

Time: 4.8s

Alternatives: 9

Speedup: N/A×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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: 98.4% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;\frac{t\_0}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\frac{{\cos x}^{2}}{c\_m} - \frac{{\sin x}^{2}}{c\_m}}{s\_m \cdot x}}{\left(s\_m \cdot x\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(\left(s\_m \cdot c\_m\right) \cdot x\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (if (<= (/ t_0 (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x))) 0.0)
     (/
      (/ (- (/ (pow (cos x) 2.0) c_m) (/ (pow (sin x) 2.0) c_m)) (* s_m x))
      (* (* s_m x) c_m))
     (/ t_0 (pow (* (* s_m c_m) x) 2.0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if ((t_0 / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= 0.0) {
		tmp = (((pow(cos(x), 2.0) / c_m) - (pow(sin(x), 2.0) / c_m)) / (s_m * x)) / ((s_m * x) * c_m);
	} else {
		tmp = t_0 / pow(((s_m * c_m) * x), 2.0);
	}
	return tmp;
}

Fortran

c_m =     private
s_m =     private
NOTE: x, 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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if ((t_0 / ((c_m ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= 0.0d0) then
        tmp = ((((cos(x) ** 2.0d0) / c_m) - ((sin(x) ** 2.0d0) / c_m)) / (s_m * x)) / ((s_m * x) * c_m)
    else
        tmp = t_0 / (((s_m * c_m) * x) ** 2.0d0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double tmp;
	if ((t_0 / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= 0.0) {
		tmp = (((Math.pow(Math.cos(x), 2.0) / c_m) - (Math.pow(Math.sin(x), 2.0) / c_m)) / (s_m * x)) / ((s_m * x) * c_m);
	} else {
		tmp = t_0 / Math.pow(((s_m * c_m) * x), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if (t_0 / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= 0.0:
		tmp = (((math.pow(math.cos(x), 2.0) / c_m) - (math.pow(math.sin(x), 2.0) / c_m)) / (s_m * x)) / ((s_m * x) * c_m)
	else:
		tmp = t_0 / math.pow(((s_m * c_m) * x), 2.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / c$95$m), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] / N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[(N[(s$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-2 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 95.8

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

   6. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-2neg N/A

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

      14. associate-/r* N/A

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

   10. Applied rewrites 99.6%

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

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

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{1}{\left(s\_m \cdot x\right) \cdot c\_m}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{t\_0}{\frac{1}{t\_1 \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(s\_m \cdot {\left(x \cdot c\_m\right)}^{1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (/ 1.0 (* (* s_m x) c_m))))
   (if (<= x 2e+16)
     (/ t_0 (/ 1.0 (* t_1 t_1)))
     (/ t_0 (pow (* s_m (pow (* x c_m) 1.0)) 2.0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double t_1 = 1.0 / ((s_m * x) * c_m);
	double tmp;
	if (x <= 2e+16) {
		tmp = t_0 / (1.0 / (t_1 * t_1));
	} else {
		tmp = t_0 / pow((s_m * pow((x * c_m), 1.0)), 2.0);
	}
	return tmp;
}

Fortran

c_m =     private
s_m =     private
NOTE: x, 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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    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))
    t_1 = 1.0d0 / ((s_m * x) * c_m)
    if (x <= 2d+16) then
        tmp = t_0 / (1.0d0 / (t_1 * t_1))
    else
        tmp = t_0 / ((s_m * ((x * c_m) ** 1.0d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double t_1 = 1.0 / ((s_m * x) * c_m);
	double tmp;
	if (x <= 2e+16) {
		tmp = t_0 / (1.0 / (t_1 * t_1));
	} else {
		tmp = t_0 / Math.pow((s_m * Math.pow((x * c_m), 1.0)), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	t_1 = 1.0 / ((s_m * x) * c_m)
	tmp = 0
	if x <= 2e+16:
		tmp = t_0 / (1.0 / (t_1 * t_1))
	else:
		tmp = t_0 / math.pow((s_m * math.pow((x * c_m), 1.0)), 2.0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(1.0 / Float64(Float64(s_m * x) * c_m))
	tmp = 0.0
	if (x <= 2e+16)
		tmp = Float64(t_0 / Float64(1.0 / Float64(t_1 * t_1)));
	else
		tmp = Float64(t_0 / (Float64(s_m * (Float64(x * c_m) ^ 1.0)) ^ 2.0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((2.0 * x));
	t_1 = 1.0 / ((s_m * x) * c_m);
	tmp = 0.0;
	if (x <= 2e+16)
		tmp = t_0 / (1.0 / (t_1 * t_1));
	else
		tmp = t_0 / ((s_m * ((x * c_m) ^ 1.0)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+16], N[(t$95$0 / N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[(s$95$m * N[Power[N[(x * c$95$m), $MachinePrecision], 1.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e16

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-neg N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. *-commutative N/A

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

      12. unpow-prod-down N/A

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

      13. pow-prod-down N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. pow-prod-down N/A

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

      18. unpow-prod-down N/A

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

      19. unpow2 N/A

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

      20. times-frac N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 97.6%

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

   if 2e16 < x

   1. Initial program 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. unpow1 N/A

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

      7. associate-*l* N/A

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

      8. unpow-prod-down N/A

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

      9. unpow1 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

Alternative 3: 96.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.9

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

4. Applied rewrites 97.9%

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

Alternative 4: 96.6% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{\left(s\_m \cdot x\right) \cdot c\_m}\\ \frac{\cos \left(2 \cdot x\right)}{\frac{1}{t\_0 \cdot t\_0}} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (* s_m x) c_m))))
   (/ (cos (* 2.0 x)) (/ 1.0 (* t_0 t_0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = 1.0 / ((s_m * x) * c_m);
	return cos((2.0 * x)) / (1.0 / (t_0 * t_0));
}

Fortran

c_m =     private
s_m =     private
NOTE: x, 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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = 1.0d0 / ((s_m * x) * c_m)
    code = cos((2.0d0 * x)) / (1.0d0 / (t_0 * t_0))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = 1.0 / ((s_m * x) * c_m);
	return Math.cos((2.0 * x)) / (1.0 / (t_0 * t_0));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = 1.0 / ((s_m * x) * c_m)
	return math.cos((2.0 * x)) / (1.0 / (t_0 * t_0))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(1.0 / Float64(Float64(s_m * x) * c_m))
	return Float64(cos(Float64(2.0 * x)) / Float64(1.0 / Float64(t_0 * t_0)))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = 1.0 / ((s_m * x) * c_m);
	tmp = cos((2.0 * x)) / (1.0 / (t_0 * t_0));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(1.0 / N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-neg N/A

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

   9. metadata-eval N/A

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

   10. pow-flip N/A

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

   11. *-commutative N/A

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

   12. unpow-prod-down N/A

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

   13. pow-prod-down N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. pow-prod-down N/A

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

   18. unpow-prod-down N/A

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

   19. unpow2 N/A

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

   20. times-frac N/A

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

   21. lower-*.f64 N/A

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

6. Applied rewrites 97.4%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\frac{1}{\frac{1}{\left(s \cdot x\right) \cdot c} \cdot \frac{1}{\left(s \cdot x\right) \cdot c}}}} \]
7. Add Preprocessing

Alternative 5: 92.6% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos \left(2 \cdot x\right)\\ t_3 := \frac{t\_2}{\left(\left(\left(\left(s\_m \cdot s\_m\right) \cdot x\right) \cdot x\right) \cdot c\_m\right) \cdot c\_m}\\ t_4 := \frac{t\_2}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)}\\ t_5 := \frac{{\cos x}^{2}}{t\_1} - \frac{{\sin x}^{2}}{t\_1}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+243}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x) c_m))
        (t_1 (* t_0 t_0))
        (t_2 (cos (* 2.0 x)))
        (t_3 (/ t_2 (* (* (* (* (* s_m s_m) x) x) c_m) c_m)))
        (t_4 (/ t_2 (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x))))
        (t_5 (- (/ (pow (cos x) 2.0) t_1) (/ (pow (sin x) 2.0) t_1))))
   (if (<= t_4 (- INFINITY))
     t_3
     (if (<= t_4 4e+243) t_5 (if (<= t_4 INFINITY) t_3 t_5)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double t_1 = t_0 * t_0;
	double t_2 = cos((2.0 * x));
	double t_3 = t_2 / (((((s_m * s_m) * x) * x) * c_m) * c_m);
	double t_4 = t_2 / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x));
	double t_5 = (pow(cos(x), 2.0) / t_1) - (pow(sin(x), 2.0) / t_1);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_4 <= 4e+243) {
		tmp = t_5;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double t_1 = t_0 * t_0;
	double t_2 = Math.cos((2.0 * x));
	double t_3 = t_2 / (((((s_m * s_m) * x) * x) * c_m) * c_m);
	double t_4 = t_2 / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x));
	double t_5 = (Math.pow(Math.cos(x), 2.0) / t_1) - (Math.pow(Math.sin(x), 2.0) / t_1);
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_4 <= 4e+243) {
		tmp = t_5;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = (s_m * x) * c_m
	t_1 = t_0 * t_0
	t_2 = math.cos((2.0 * x))
	t_3 = t_2 / (((((s_m * s_m) * x) * x) * c_m) * c_m)
	t_4 = t_2 / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))
	t_5 = (math.pow(math.cos(x), 2.0) / t_1) - (math.pow(math.sin(x), 2.0) / t_1)
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_3
	elif t_4 <= 4e+243:
		tmp = t_5
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_5
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(s_m * x) * c_m)
	t_1 = Float64(t_0 * t_0)
	t_2 = cos(Float64(2.0 * x))
	t_3 = Float64(t_2 / Float64(Float64(Float64(Float64(Float64(s_m * s_m) * x) * x) * c_m) * c_m))
	t_4 = Float64(t_2 / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x)))
	t_5 = Float64(Float64((cos(x) ^ 2.0) / t_1) - Float64((sin(x) ^ 2.0) / t_1))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_4 <= 4e+243)
		tmp = t_5;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = (s_m * x) * c_m;
	t_1 = t_0 * t_0;
	t_2 = cos((2.0 * x));
	t_3 = t_2 / (((((s_m * s_m) * x) * x) * c_m) * c_m);
	t_4 = t_2 / ((c_m ^ 2.0) * ((x * (s_m ^ 2.0)) * x));
	t_5 = ((cos(x) ^ 2.0) / t_1) - ((sin(x) ^ 2.0) / t_1);
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_3;
	elseif (t_4 <= 4e+243)
		tmp = t_5;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$3, If[LessEqual[t$95$4, 4e+243], t$95$5, If[LessEqual[t$95$4, Infinity], t$95$3, t$95$5]]]]]]]]]

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))) < -inf.0 or 4.0000000000000003e243 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   if -inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 4.0000000000000003e243 or +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   6. Applied rewrites 89.2%

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

Alternative 6: 89.9% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\_m\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos \left(2 \cdot x\right)\\ t_3 := \frac{t\_2}{\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\right) \cdot \left(s\_m \cdot s\_m\right)\right) \cdot x}\\ t_4 := \frac{t\_2}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)}\\ t_5 := \frac{{\cos x}^{2}}{t\_1} - \frac{{\sin x}^{2}}{t\_1}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x) c_m))
        (t_1 (* t_0 t_0))
        (t_2 (cos (* 2.0 x)))
        (t_3 (/ t_2 (* (* (* (* c_m c_m) x) (* s_m s_m)) x)))
        (t_4 (/ t_2 (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x))))
        (t_5 (- (/ (pow (cos x) 2.0) t_1) (/ (pow (sin x) 2.0) t_1))))
   (if (<= t_4 (- INFINITY))
     t_3
     (if (<= t_4 5e+283) t_5 (if (<= t_4 INFINITY) t_3 t_5)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double t_1 = t_0 * t_0;
	double t_2 = cos((2.0 * x));
	double t_3 = t_2 / ((((c_m * c_m) * x) * (s_m * s_m)) * x);
	double t_4 = t_2 / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x));
	double t_5 = (pow(cos(x), 2.0) / t_1) - (pow(sin(x), 2.0) / t_1);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_4 <= 5e+283) {
		tmp = t_5;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = (s_m * x) * c_m;
	double t_1 = t_0 * t_0;
	double t_2 = Math.cos((2.0 * x));
	double t_3 = t_2 / ((((c_m * c_m) * x) * (s_m * s_m)) * x);
	double t_4 = t_2 / (Math.pow(c_m, 2.0) * ((x * Math.pow(s_m, 2.0)) * x));
	double t_5 = (Math.pow(Math.cos(x), 2.0) / t_1) - (Math.pow(Math.sin(x), 2.0) / t_1);
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_4 <= 5e+283) {
		tmp = t_5;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = (s_m * x) * c_m
	t_1 = t_0 * t_0
	t_2 = math.cos((2.0 * x))
	t_3 = t_2 / ((((c_m * c_m) * x) * (s_m * s_m)) * x)
	t_4 = t_2 / (math.pow(c_m, 2.0) * ((x * math.pow(s_m, 2.0)) * x))
	t_5 = (math.pow(math.cos(x), 2.0) / t_1) - (math.pow(math.sin(x), 2.0) / t_1)
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_3
	elif t_4 <= 5e+283:
		tmp = t_5
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = t_5
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(s_m * x) * c_m)
	t_1 = Float64(t_0 * t_0)
	t_2 = cos(Float64(2.0 * x))
	t_3 = Float64(t_2 / Float64(Float64(Float64(Float64(c_m * c_m) * x) * Float64(s_m * s_m)) * x))
	t_4 = Float64(t_2 / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x)))
	t_5 = Float64(Float64((cos(x) ^ 2.0) / t_1) - Float64((sin(x) ^ 2.0) / t_1))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_4 <= 5e+283)
		tmp = t_5;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = (s_m * x) * c_m;
	t_1 = t_0 * t_0;
	t_2 = cos((2.0 * x));
	t_3 = t_2 / ((((c_m * c_m) * x) * (s_m * s_m)) * x);
	t_4 = t_2 / ((c_m ^ 2.0) * ((x * (s_m ^ 2.0)) * x));
	t_5 = ((cos(x) ^ 2.0) / t_1) - ((sin(x) ^ 2.0) / t_1);
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_3;
	elseif (t_4 <= 5e+283)
		tmp = t_5;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$3, If[LessEqual[t$95$4, 5e+283], t$95$5, If[LessEqual[t$95$4, Infinity], t$95$3, t$95$5]]]]]]]]]

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))) < -inf.0 or 5.0000000000000004e283 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 5.0000000000000004e283 or +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   6. Applied rewrites 89.3%

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

Alternative 7: 85.2% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)}\\ t_1 := \left(s\_m \cdot x\right) \cdot c\_m\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \frac{{\cos x}^{2}}{t\_2} - \frac{{\sin x}^{2}}{t\_2}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+243}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\sin \left(\mathsf{fma}\left(x, 2, \frac{\pi}{2}\right)\right)}{{\left(c\_m \cdot s\_m\right)}^{2}}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s_m 2.0)) x))))
        (t_1 (* (* s_m x) c_m))
        (t_2 (* t_1 t_1))
        (t_3 (- (/ (pow (cos x) 2.0) t_2) (/ (pow (sin x) 2.0) t_2))))
   (if (<= t_0 4e+243)
     t_3
     (if (<= t_0 INFINITY)
       (/ (/ (sin (fma x 2.0 (/ PI 2.0))) (pow (* c_m s_m) 2.0)) (* x x))
       t_3))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s_m, 2.0)) * x));
	double t_1 = (s_m * x) * c_m;
	double t_2 = t_1 * t_1;
	double t_3 = (pow(cos(x), 2.0) / t_2) - (pow(sin(x), 2.0) / t_2);
	double tmp;
	if (t_0 <= 4e+243) {
		tmp = t_3;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sin(fma(x, 2.0, (((double) M_PI) / 2.0))) / pow((c_m * s_m), 2.0)) / (x * x);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x)))
	t_1 = Float64(Float64(s_m * x) * c_m)
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(Float64((cos(x) ^ 2.0) / t_2) - Float64((sin(x) ^ 2.0) / t_2))
	tmp = 0.0
	if (t_0 <= 4e+243)
		tmp = t_3;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sin(fma(x, 2.0, Float64(pi / 2.0))) / (Float64(c_m * s_m) ^ 2.0)) / Float64(x * x));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+243], t$95$3, If[LessEqual[t$95$0, Infinity], N[(N[(N[Sin[N[(x * 2.0 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * s$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 4.0000000000000003e243 or +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

   6. Applied rewrites 84.2%

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

   if 4.0000000000000003e243 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. div-sub N/A

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.9%

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

Alternative 8: 71.2% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := {\left(c\_m \cdot x\right)}^{1}\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\sin \left(\mathsf{fma}\left(x, 2, \frac{\pi}{2}\right)\right)}{{\left(c\_m \cdot s\_m\right)}^{2}}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\cos x}^{2}}{t\_1}}{s\_m \cdot s\_m} - \frac{\frac{{\sin x}^{2}}{t\_1}}{s\_m \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (pow (* c_m x) 1.0)) (t_1 (* t_0 t_0)))
   (if (<= x 2.7e+93)
     (/ (/ (sin (fma x 2.0 (/ PI 2.0))) (pow (* c_m s_m) 2.0)) (* x x))
     (-
      (/ (/ (pow (cos x) 2.0) t_1) (* s_m s_m))
      (/ (/ (pow (sin x) 2.0) t_1) (* s_m s_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = pow((c_m * x), 1.0);
	double t_1 = t_0 * t_0;
	double tmp;
	if (x <= 2.7e+93) {
		tmp = (sin(fma(x, 2.0, (((double) M_PI) / 2.0))) / pow((c_m * s_m), 2.0)) / (x * x);
	} else {
		tmp = ((pow(cos(x), 2.0) / t_1) / (s_m * s_m)) - ((pow(sin(x), 2.0) / t_1) / (s_m * s_m));
	}
	return tmp;
}

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * x) ^ 1.0
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (x <= 2.7e+93)
		tmp = Float64(Float64(sin(fma(x, 2.0, Float64(pi / 2.0))) / (Float64(c_m * s_m) ^ 2.0)) / Float64(x * x));
	else
		tmp = Float64(Float64(Float64((cos(x) ^ 2.0) / t_1) / Float64(s_m * s_m)) - Float64(Float64((sin(x) ^ 2.0) / t_1) / Float64(s_m * s_m)));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Power[N[(c$95$m * x), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[x, 2.7e+93], N[(N[(N[Sin[N[(x * 2.0 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * s$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6999999999999999e93

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. div-sub N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.0%

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

   if 2.6999999999999999e93 < x

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. div-sub N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in s around 0

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

   7. Applied rewrites 69.3%

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

Alternative 9: 48.8% accurate, N/A× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := {\left(c\_m \cdot x\right)}^{1}\\ t_1 := t\_0 \cdot t\_0\\ \frac{\frac{{\cos x}^{2}}{t\_1}}{s\_m \cdot s\_m} - \frac{\frac{{\sin x}^{2}}{t\_1}}{s\_m \cdot s\_m} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (pow (* c_m x) 1.0)) (t_1 (* t_0 t_0)))
   (-
    (/ (/ (pow (cos x) 2.0) t_1) (* s_m s_m))
    (/ (/ (pow (sin x) 2.0) t_1) (* s_m s_m)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = pow((c_m * x), 1.0);
	double t_1 = t_0 * t_0;
	return ((pow(cos(x), 2.0) / t_1) / (s_m * s_m)) - ((pow(sin(x), 2.0) / t_1) / (s_m * s_m));
}

Fortran

c_m =     private
s_m =     private
NOTE: x, 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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (c_m * x) ** 1.0d0
    t_1 = t_0 * t_0
    code = (((cos(x) ** 2.0d0) / t_1) / (s_m * s_m)) - (((sin(x) ** 2.0d0) / t_1) / (s_m * s_m))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.pow((c_m * x), 1.0);
	double t_1 = t_0 * t_0;
	return ((Math.pow(Math.cos(x), 2.0) / t_1) / (s_m * s_m)) - ((Math.pow(Math.sin(x), 2.0) / t_1) / (s_m * s_m));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.pow((c_m * x), 1.0)
	t_1 = t_0 * t_0
	return ((math.pow(math.cos(x), 2.0) / t_1) / (s_m * s_m)) - ((math.pow(math.sin(x), 2.0) / t_1) / (s_m * s_m))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * x) ^ 1.0
	t_1 = Float64(t_0 * t_0)
	return Float64(Float64(Float64((cos(x) ^ 2.0) / t_1) / Float64(s_m * s_m)) - Float64(Float64((sin(x) ^ 2.0) / t_1) / Float64(s_m * s_m)))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = (c_m * x) ^ 1.0;
	t_1 = t_0 * t_0;
	tmp = (((cos(x) ^ 2.0) / t_1) / (s_m * s_m)) - (((sin(x) ^ 2.0) / t_1) / (s_m * s_m));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Power[N[(c$95$m * x), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, N[(N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. div-sub N/A

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

4. Applied rewrites 42.9%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\frac{\frac{{\cos x}^{2}}{{\left(c \cdot x\right)}^{1} \cdot {\left(c \cdot x\right)}^{1}}}{s \cdot s} - \frac{\frac{{\sin x}^{2}}{{\left(c \cdot x\right)}^{1} \cdot {\left(c \cdot x\right)}^{1}}}{s \cdot s}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))