mixedcos

Percentage Accurate: 66.3% → 98.5%

Time: 4.7s

Alternatives: 10

Speedup: 9.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{-59}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{elif}\;x\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(x\_m \cdot x\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(c\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x_m))))
   (if (<= x_m 9e-59)
     (/ t_0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))
     (if (<= x_m 1.32e+154)
       (/ (cos (+ x_m x_m)) (* (* (* x_m x_m) (* c_m s_m)) (* c_m s_m)))
       (/ t_0 (* (* (* s_m s_m) (* c_m x_m)) (* c_m x_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((2.0 * x_m));
	double tmp;
	if (x_m <= 9e-59) {
		tmp = t_0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else if (x_m <= 1.32e+154) {
		tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x_m))
    if (x_m <= 9d-59) then
        tmp = t_0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    else if (x_m <= 1.32d+154) then
        tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m))
    else
        tmp = t_0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x_m));
	double tmp;
	if (x_m <= 9e-59) {
		tmp = t_0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else if (x_m <= 1.32e+154) {
		tmp = Math.cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((2.0 * x_m))
	tmp = 0
	if x_m <= 9e-59:
		tmp = t_0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	elif x_m <= 1.32e+154:
		tmp = math.cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m))
	else:
		tmp = t_0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(2.0 * x_m))
	tmp = 0.0
	if (x_m <= 9e-59)
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	elseif (x_m <= 1.32e+154)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(x_m * x_m) * Float64(c_m * s_m)) * Float64(c_m * s_m)));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x_m)) * Float64(c_m * x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((2.0 * x_m));
	tmp = 0.0;
	if (x_m <= 9e-59)
		tmp = t_0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	elseif (x_m <= 1.32e+154)
		tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	else
		tmp = t_0 / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 9e-59], N[(t$95$0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.32e+154], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.00000000000000023e-59

   1. Initial program 64.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

   if 9.00000000000000023e-59 < x < 1.31999999999999998e154

   1. Initial program 66.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

   if 1.31999999999999998e154 < x

   1. Initial program 54.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. swap-sqr N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.3

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

   6. Applied rewrites 67.3%

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

Alternative 2: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -1 \cdot 10^{-265}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -1e-265)
     (/ (cos (+ x_m x_m)) (* (* (* c_m c_m) (* x_m x_m)) (* s_m s_m)))
     (/ (/ 1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e-265) {
		tmp = cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-1d-265)) then
        tmp = cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -1e-265) {
		tmp = Math.cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -1e-265:
		tmp = math.cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e-265)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(c_m * c_m) * Float64(x_m * x_m)) * Float64(s_m * s_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -1e-265)
		tmp = cos((x_m + x_m)) / (((c_m * c_m) * (x_m * x_m)) * (s_m * s_m));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-265], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.99999999999999985e-266

   1. Initial program 65.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.8

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

   4. Applied rewrites 32.8%

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

      1. count-2-rev N/A

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

      2. lower-+.f64 32.8

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

   6. Applied rewrites 32.8%

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

   if -9.99999999999999985e-266 < (/.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 63.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 58.4

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

      2. distribute-lft-neg-in 58.4

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

      3. metadata-eval 58.4

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

   7. Applied rewrites 58.4%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 67.4

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

   9. Applied rewrites 67.4%

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. unpow-prod-down N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 81.8%

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

Alternative 3: 82.9% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -1e+153)
     (/ (fma (* x_m x_m) -2.0 1.0) (* (* (* (* s_m x_m) c_m) x_m) (* c_m s_m)))
     (/ (/ 1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e+153) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((((s_m * x_m) * c_m) * x_m) * (c_m * s_m));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e+153)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * x_m) * Float64(c_m * s_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+153], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e153

   1. Initial program 52.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.9

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in x around 0

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

      1. cos-neg-rev N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.8

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

   9. Applied rewrites 36.8%

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

   if -1e153 < (/.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 64.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 57.0

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

      2. distribute-lft-neg-in 57.0

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

      3. metadata-eval 57.0

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

   7. Applied rewrites 57.0%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 65.8

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

   9. Applied rewrites 65.8%

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. unpow-prod-down N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 79.9%

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

Alternative 4: 98.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{elif}\;x\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(x\_m \cdot x\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(c\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 9.6e-11)
     (/ (/ 1.0 t_0) t_0)
     (if (<= x_m 1.32e+154)
       (/ (cos (+ x_m x_m)) (* (* (* x_m x_m) (* c_m s_m)) (* c_m s_m)))
       (/ (cos (* 2.0 x_m)) (* (* (* s_m s_m) (* c_m x_m)) (* c_m x_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 9.6e-11) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x_m <= 1.32e+154) {
		tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	} else {
		tmp = cos((2.0 * x_m)) / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 9.6d-11) then
        tmp = (1.0d0 / t_0) / t_0
    else if (x_m <= 1.32d+154) then
        tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m))
    else
        tmp = cos((2.0d0 * x_m)) / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 9.6e-11) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x_m <= 1.32e+154) {
		tmp = Math.cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	} else {
		tmp = Math.cos((2.0 * x_m)) / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 9.6e-11:
		tmp = (1.0 / t_0) / t_0
	elif x_m <= 1.32e+154:
		tmp = math.cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m))
	else:
		tmp = math.cos((2.0 * x_m)) / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 9.6e-11)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	elseif (x_m <= 1.32e+154)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(x_m * x_m) * Float64(c_m * s_m)) * Float64(c_m * s_m)));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x_m)) * Float64(c_m * x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 9.6e-11)
		tmp = (1.0 / t_0) / t_0;
	elseif (x_m <= 1.32e+154)
		tmp = cos((x_m + x_m)) / (((x_m * x_m) * (c_m * s_m)) * (c_m * s_m));
	else
		tmp = cos((2.0 * x_m)) / (((s_m * s_m) * (c_m * x_m)) * (c_m * x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 9.6e-11], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$95$m, 1.32e+154], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.6000000000000005e-11

   1. Initial program 64.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 58.1

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 55.2

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

      2. distribute-lft-neg-in 55.2

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

      3. metadata-eval 55.2

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

   7. Applied rewrites 55.2%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. unpow-prod-down N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 80.5%

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

   if 9.6000000000000005e-11 < x < 1.31999999999999998e154

   1. Initial program 70.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

      1. count-2-rev N/A

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

      2. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

   if 1.31999999999999998e154 < x

   1. Initial program 54.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. swap-sqr N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.3

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

   6. Applied rewrites 67.3%

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

Alternative 5: 94.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(c\_m \cdot x\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 3.5e-13)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x_m x_m)) (* (* (* c_m x_m) x_m) (* s_m (* s_m c_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 3.5e-13) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m + x_m)) / (((c_m * x_m) * x_m) * (s_m * (s_m * c_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 3.5d-13) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m + x_m)) / (((c_m * x_m) * x_m) * (s_m * (s_m * c_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 3.5e-13) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m + x_m)) / (((c_m * x_m) * x_m) * (s_m * (s_m * c_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 3.5e-13:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m + x_m)) / (((c_m * x_m) * x_m) * (s_m * (s_m * c_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 3.5e-13)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(c_m * x_m) * x_m) * Float64(s_m * Float64(s_m * c_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 3.5e-13)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m + x_m)) / (((c_m * x_m) * x_m) * (s_m * (s_m * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.5e-13], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5000000000000002e-13

   1. Initial program 63.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 57.6

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 54.7

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

      2. distribute-lft-neg-in 54.7

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

      3. metadata-eval 54.7

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

   7. Applied rewrites 54.7%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.6

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

   9. Applied rewrites 64.6%

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. unpow-prod-down N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 80.3%

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

   if 3.5000000000000002e-13 < x

   1. Initial program 64.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

      1. unpow2 N/A

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

      2. swap-sqr N/A

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

      3. sqr-neg-rev N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 97.5

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

   6. Applied rewrites 97.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

      1. count-2-rev N/A

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

      2. lower-+.f64 80.7

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

   10. Applied rewrites 80.7%

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

Alternative 6: 97.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. pow-prod-down N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. 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. unpow2 N/A

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

   2. swap-sqr N/A

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

   3. sqr-neg-rev N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 97.9

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

6. Applied rewrites 97.9%

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

7. Final simplification 97.9%

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

Alternative 7: 71.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.76 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(x\_m \cdot \left(\left(\left(c\_m \cdot s\_m\right) \cdot s\_m\right) \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.76e-159)
   (/ 1.0 (* c_m (* x_m (* (* (* c_m s_m) s_m) x_m))))
   (/ 1.0 (* (* c_m s_m) (* (* c_m s_m) (* x_m x_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.76e-159) {
		tmp = 1.0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)));
	} else {
		tmp = 1.0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.76d-159) then
        tmp = 1.0d0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)))
    else
        tmp = 1.0d0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.76e-159) {
		tmp = 1.0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)));
	} else {
		tmp = 1.0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.76e-159:
		tmp = 1.0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)))
	else:
		tmp = 1.0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.76e-159)
		tmp = Float64(1.0 / Float64(c_m * Float64(x_m * Float64(Float64(Float64(c_m * s_m) * s_m) * x_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(Float64(c_m * s_m) * Float64(x_m * x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.76e-159)
		tmp = 1.0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)));
	else
		tmp = 1.0 / ((c_m * s_m) * ((c_m * s_m) * (x_m * x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.76e-159], N[(1.0 / N[(c$95$m * N[(x$95$m * N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.76e-159

   1. Initial program 64.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 57.5

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 54.2

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

      2. distribute-lft-neg-in 54.2

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

      3. metadata-eval 54.2

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

   7. Applied rewrites 54.2%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

       1. pow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 69.9

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

   11. Applied rewrites 69.9%

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

   if 1.76e-159 < x

   1. Initial program 62.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

      1. cos-neg-rev 52.8

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

      2. distribute-lft-neg-in 52.8

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

      3. metadata-eval 52.8

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

   7. Applied rewrites 52.8%

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

      1. unswap-sqr N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 56.6

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

   9. Applied rewrites 56.6%

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

       1. unswap-sqr N/A

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

       2. pow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. unpow-prod-down N/A

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

       7. pow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lower-*.f64 69.9

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

   11. Applied rewrites 69.9%

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

Alternative 8: 79.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x_m)
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 57.8

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in x around 0

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

   1. cos-neg-rev 53.7

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

   2. distribute-lft-neg-in 53.7

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

   3. metadata-eval 53.7

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

7. Applied rewrites 53.7%

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

   1. unswap-sqr N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 61.9

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

9. Applied rewrites 61.9%

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

    1. pow2 N/A

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

    2. *-commutative N/A

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

    3. pow2 N/A

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

    4. unpow-prod-down N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. unpow2 N/A

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

    8. associate-/r* N/A

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

    9. lower-/.f64 N/A

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

11. Applied rewrites 75.3%

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

Alternative 9: 79.1% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 57.8

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in x around 0

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

   1. cos-neg-rev 53.7

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

   2. distribute-lft-neg-in 53.7

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

   3. metadata-eval 53.7

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

7. Applied rewrites 53.7%

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

   1. unswap-sqr N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 61.9

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

9. Applied rewrites 61.9%

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

    1. unswap-sqr N/A

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

    2. pow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. *-commutative N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lower-*.f64 75.3

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

11. Applied rewrites 75.3%

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

Alternative 10: 69.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(x\_m \cdot \left(\left(\left(c\_m \cdot s\_m\right) \cdot s\_m\right) \cdot x\_m\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * (x_m * (((c_m * s_m) * s_m) * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 57.8

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in x around 0

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

   1. cos-neg-rev 53.7

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

   2. distribute-lft-neg-in 53.7

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

   3. metadata-eval 53.7

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

7. Applied rewrites 53.7%

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

   1. unswap-sqr N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 61.9

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

9. Applied rewrites 61.9%

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

    1. pow2 N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. pow2 N/A

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

    7. associate-*r* N/A

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

    8. associate-*l* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. *-commutative N/A

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

    14. *-commutative N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 67.2

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

11. Applied rewrites 67.2%

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

Reproduce

herbie shell --seed 2025044 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))