mixedcos

Percentage Accurate: 66.3% → 99.2%

Time: 4.1s

Alternatives: 9

Speedup: 4.2×

• 69.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \cos \left(\left|x\right| + \left|x\right|\right)\\ t_2 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := \left(t\_2 \cdot \left|x\right|\right) \cdot t\_0\\ \mathbf{if}\;\left|x\right| \leq 10^{+47}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_3}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{t\_0} \cdot \frac{1}{\left|x\right|}}{\left(t\_0 \cdot \left|x\right|\right) \cdot t\_2}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmin (fabs c) (fabs s)))
        (t_1 (cos (+ (fabs x) (fabs x))))
        (t_2 (fmax (fabs c) (fabs s)))
        (t_3 (* (* t_2 (fabs x)) t_0)))
   (if (<= (fabs x) 1e+47)
     (/ (/ t_1 t_3) t_3)
     (/ (/ (* (/ t_1 t_0) (/ 1.0 (fabs x))) (* (* t_0 (fabs x)) t_2)) t_2))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = cos((fabs(x) + fabs(x)));
	double t_2 = fmax(fabs(c), fabs(s));
	double t_3 = (t_2 * fabs(x)) * t_0;
	double tmp;
	if (fabs(x) <= 1e+47) {
		tmp = (t_1 / t_3) / t_3;
	} else {
		tmp = (((t_1 / t_0) * (1.0 / fabs(x))) / ((t_0 * fabs(x)) * t_2)) / t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = cos((abs(x) + abs(x)))
    t_2 = fmax(abs(c), abs(s))
    t_3 = (t_2 * abs(x)) * t_0
    if (abs(x) <= 1d+47) then
        tmp = (t_1 / t_3) / t_3
    else
        tmp = (((t_1 / t_0) * (1.0d0 / abs(x))) / ((t_0 * abs(x)) * t_2)) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = Math.cos((Math.abs(x) + Math.abs(x)));
	double t_2 = fmax(Math.abs(c), Math.abs(s));
	double t_3 = (t_2 * Math.abs(x)) * t_0;
	double tmp;
	if (Math.abs(x) <= 1e+47) {
		tmp = (t_1 / t_3) / t_3;
	} else {
		tmp = (((t_1 / t_0) * (1.0 / Math.abs(x))) / ((t_0 * Math.abs(x)) * t_2)) / t_2;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = math.cos((math.fabs(x) + math.fabs(x)))
	t_2 = fmax(math.fabs(c), math.fabs(s))
	t_3 = (t_2 * math.fabs(x)) * t_0
	tmp = 0
	if math.fabs(x) <= 1e+47:
		tmp = (t_1 / t_3) / t_3
	else:
		tmp = (((t_1 / t_0) * (1.0 / math.fabs(x))) / ((t_0 * math.fabs(x)) * t_2)) / t_2
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = cos(Float64(abs(x) + abs(x)))
	t_2 = fmax(abs(c), abs(s))
	t_3 = Float64(Float64(t_2 * abs(x)) * t_0)
	tmp = 0.0
	if (abs(x) <= 1e+47)
		tmp = Float64(Float64(t_1 / t_3) / t_3);
	else
		tmp = Float64(Float64(Float64(Float64(t_1 / t_0) * Float64(1.0 / abs(x))) / Float64(Float64(t_0 * abs(x)) * t_2)) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = cos((abs(x) + abs(x)));
	t_2 = max(abs(c), abs(s));
	t_3 = (t_2 * abs(x)) * t_0;
	tmp = 0.0;
	if (abs(x) <= 1e+47)
		tmp = (t_1 / t_3) / t_3;
	else
		tmp = (((t_1 / t_0) * (1.0 / abs(x))) / ((t_0 * abs(x)) * t_2)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e+47], N[(N[(t$95$1 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e47

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7%

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

   3. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 97.1%

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

   if 1e47 < x

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-/r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. frac-times N/A

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

      18. lower-/.f64 N/A

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

   5. Applied rewrites 93.6%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, s\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := \mathsf{min}\left(\left|c\right|, s\right)\\ t_3 := \left(t\_0 \cdot x\right) \cdot t\_2\\ t_4 := \left(t\_2 \cdot t\_0\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{t\_2}^{2} \cdot \left(\left(x \cdot {t\_0}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_1}{t\_3}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_4}}{t\_4}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) s))
        (t_1 (cos (+ x x)))
        (t_2 (fmin (fabs c) s))
        (t_3 (* (* t_0 x) t_2))
        (t_4 (* (* t_2 t_0) x)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow t_2 2.0) (* (* x (pow t_0 2.0)) x)))
        INFINITY)
     (/ (/ t_1 t_3) t_3)
     (/ (/ t_1 t_4) t_4))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), s);
	double t_1 = cos((x + x));
	double t_2 = fmin(fabs(c), s);
	double t_3 = (t_0 * x) * t_2;
	double t_4 = (t_2 * t_0) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(t_2, 2.0) * ((x * pow(t_0, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_3) / t_3;
	} else {
		tmp = (t_1 / t_4) / t_4;
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), s);
	double t_1 = Math.cos((x + x));
	double t_2 = fmin(Math.abs(c), s);
	double t_3 = (t_0 * x) * t_2;
	double t_4 = (t_2 * t_0) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(t_2, 2.0) * ((x * Math.pow(t_0, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / t_3) / t_3;
	} else {
		tmp = (t_1 / t_4) / t_4;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), s)
	t_1 = math.cos((x + x))
	t_2 = fmin(math.fabs(c), s)
	t_3 = (t_0 * x) * t_2
	t_4 = (t_2 * t_0) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(t_2, 2.0) * ((x * math.pow(t_0, 2.0)) * x))) <= math.inf:
		tmp = (t_1 / t_3) / t_3
	else:
		tmp = (t_1 / t_4) / t_4
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), s)
	t_1 = cos(Float64(x + x))
	t_2 = fmin(abs(c), s)
	t_3 = Float64(Float64(t_0 * x) * t_2)
	t_4 = Float64(Float64(t_2 * t_0) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((t_2 ^ 2.0) * Float64(Float64(x * (t_0 ^ 2.0)) * x))) <= Inf)
		tmp = Float64(Float64(t_1 / t_3) / t_3);
	else
		tmp = Float64(Float64(t_1 / t_4) / t_4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), s);
	t_1 = cos((x + x));
	t_2 = min(abs(c), s);
	t_3 = (t_0 * x) * t_2;
	t_4 = (t_2 * t_0) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((t_2 ^ 2.0) * ((x * (t_0 ^ 2.0)) * x))) <= Inf)
		tmp = (t_1 / t_3) / t_3;
	else
		tmp = (t_1 / t_4) / t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * t$95$0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(x * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(t$95$1 / t$95$4), $MachinePrecision] / t$95$4), $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))) < +inf.0

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7%

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

   3. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 97.1%

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

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

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7%

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

   3. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 97.1%

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 94.9%

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

   9. Applied rewrites 94.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 97.3%

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

   11. Applied rewrites 97.3%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, s\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, s\right)\\ t_2 := \left(t\_0 \cdot t\_1\right) \cdot \left|x\right|\\ t_3 := \left(t\_1 \cdot \left|x\right|\right) \cdot t\_0\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{t\_3}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_2}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmin (fabs c) s))
        (t_1 (fmax (fabs c) s))
        (t_2 (* (* t_0 t_1) (fabs x)))
        (t_3 (* (* t_1 (fabs x)) t_0)))
   (if (<= (fabs x) 2e-121)
     (/ (/ 1.0 t_3) t_3)
     (/ (/ (cos (+ (fabs x) (fabs x))) t_2) t_2))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), s);
	double t_1 = fmax(fabs(c), s);
	double t_2 = (t_0 * t_1) * fabs(x);
	double t_3 = (t_1 * fabs(x)) * t_0;
	double tmp;
	if (fabs(x) <= 2e-121) {
		tmp = (1.0 / t_3) / t_3;
	} else {
		tmp = (cos((fabs(x) + fabs(x))) / t_2) / t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(c), s)
    t_1 = fmax(abs(c), s)
    t_2 = (t_0 * t_1) * abs(x)
    t_3 = (t_1 * abs(x)) * t_0
    if (abs(x) <= 2d-121) then
        tmp = (1.0d0 / t_3) / t_3
    else
        tmp = (cos((abs(x) + abs(x))) / t_2) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), s);
	double t_1 = fmax(Math.abs(c), s);
	double t_2 = (t_0 * t_1) * Math.abs(x);
	double t_3 = (t_1 * Math.abs(x)) * t_0;
	double tmp;
	if (Math.abs(x) <= 2e-121) {
		tmp = (1.0 / t_3) / t_3;
	} else {
		tmp = (Math.cos((Math.abs(x) + Math.abs(x))) / t_2) / t_2;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), s)
	t_1 = fmax(math.fabs(c), s)
	t_2 = (t_0 * t_1) * math.fabs(x)
	t_3 = (t_1 * math.fabs(x)) * t_0
	tmp = 0
	if math.fabs(x) <= 2e-121:
		tmp = (1.0 / t_3) / t_3
	else:
		tmp = (math.cos((math.fabs(x) + math.fabs(x))) / t_2) / t_2
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), s)
	t_1 = fmax(abs(c), s)
	t_2 = Float64(Float64(t_0 * t_1) * abs(x))
	t_3 = Float64(Float64(t_1 * abs(x)) * t_0)
	tmp = 0.0
	if (abs(x) <= 2e-121)
		tmp = Float64(Float64(1.0 / t_3) / t_3);
	else
		tmp = Float64(Float64(cos(Float64(abs(x) + abs(x))) / t_2) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), s);
	t_1 = max(abs(c), s);
	t_2 = (t_0 * t_1) * abs(x);
	t_3 = (t_1 * abs(x)) * t_0;
	tmp = 0.0;
	if (abs(x) <= 2e-121)
		tmp = (1.0 / t_3) / t_3;
	else
		tmp = (cos((abs(x) + abs(x))) / t_2) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-121], N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-121

   1. Initial program 66.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. pow-prod-down N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

   8. Applied rewrites 79.3%

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

   if 2e-121 < x

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7%

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

   3. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 97.1%

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 94.9%

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

   9. Applied rewrites 94.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 97.3%

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

   11. Applied rewrites 97.3%

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

Alternative 4: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\mathsf{max}\left(c, s\right) \cdot x\right) \cdot \mathsf{min}\left(c, s\right)\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* (fmax c s) x) (fmin c s)))) (/ (cos (+ x x)) (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (fmax(c, s) * x) * fmin(c, s);
	return cos((x + x)) / (t_0 * t_0);
}

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
    real(8) :: t_0
    t_0 = (fmax(c, s) * x) * fmin(c, s)
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (fmax(c, s) * x) * fmin(c, s);
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = (fmax(c, s) * x) * fmin(c, s)
	return math.cos((x + x)) / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(fmax(c, s) * x) * fmin(c, s))
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (max(c, s) * x) * min(c, s);
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(N[Max[c, s], $MachinePrecision] * x), $MachinePrecision] * N[Min[c, s], $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 76.7%

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

3. Applied rewrites 76.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 96.9%

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

5. Applied rewrites 96.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 96.9%

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

7. Applied rewrites 96.9%

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

Alternative 5: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\mathsf{max}\left(c, s\right) \cdot x\right) \cdot \mathsf{min}\left(c, s\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{\left(\mathsf{min}\left(c, s\right)\right)}^{2} \cdot \left(\left(x \cdot {\left(\mathsf{max}\left(c, s\right)\right)}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\mathsf{max}\left(c, s\right) \cdot \left(\left(\mathsf{min}\left(c, s\right) \cdot x\right) \cdot x\right)\right) \cdot \left(\mathsf{min}\left(c, s\right) \cdot \mathsf{max}\left(c, s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* (fmax c s) x) (fmin c s))))
   (if (<=
        (/
         (cos (* 2.0 x))
         (* (pow (fmin c s) 2.0) (* (* x (pow (fmax c s) 2.0)) x)))
        -1e-127)
     (/
      (fma (* x x) -2.0 1.0)
      (* (* (fmax c s) (* (* (fmin c s) x) x)) (* (fmin c s) (fmax c s))))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (fmax(c, s) * x) * fmin(c, s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(fmin(c, s), 2.0) * ((x * pow(fmax(c, s), 2.0)) * x))) <= -1e-127) {
		tmp = fma((x * x), -2.0, 1.0) / ((fmax(c, s) * ((fmin(c, s) * x) * x)) * (fmin(c, s) * fmax(c, s)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(fmax(c, s) * x) * fmin(c, s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((fmin(c, s) ^ 2.0) * Float64(Float64(x * (fmax(c, s) ^ 2.0)) * x))) <= -1e-127)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(fmax(c, s) * Float64(Float64(fmin(c, s) * x) * x)) * Float64(fmin(c, s) * fmax(c, s))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(N[Max[c, s], $MachinePrecision] * x), $MachinePrecision] * N[Min[c, s], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[N[Min[c, s], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(x * N[Power[N[Max[c, s], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-127], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Max[c, s], $MachinePrecision] * N[(N[(N[Min[c, s], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[Min[c, s], $MachinePrecision] * N[Max[c, s], $MachinePrecision]), $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))) < -1e-127

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7%

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

   3. Applied rewrites 76.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.2%

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

   6. Applied rewrites 51.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 51.2%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.2%

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

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left({c}^{2} \cdot \left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot {c}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)\right) \cdot {c}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \mathsf{Rewrite=>}\left(lift-pow.f64, \left({c}^{2}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \mathsf{Rewrite<=}\left(pow2, \left(c \cdot c\right)\right)} \]

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(\left(s \cdot x\right) \cdot c\right)\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(s \cdot x\right)\right) \cdot c\right)} \]

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \mathsf{Rewrite=>}\left(associate-*l*, \left(x \cdot \left(s \cdot c\right)\right)\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(x \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(s \cdot c\right)\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(associate-*r*, \left(\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)\right)\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)\right)\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right)\right) \cdot \left(s \cdot c\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(s \cdot c\right)\right)} \]

      24. lower-*.f64 N/A

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

      25. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right) \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(c \cdot s\right)\right)} \]

   8. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.8%

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

   10. Applied rewrites 55.8%

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

   if -1e-127 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. pow-prod-down N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

   8. Applied rewrites 79.3%

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

Alternative 6: 79.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\mathsf{max}\left(c, s\right) \cdot x\right) \cdot \mathsf{min}\left(c, s\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* (fmax c s) x) (fmin c s)))) (/ (/ 1.0 t_0) t_0)))

C

double code(double x, double c, double s) {
	double t_0 = (fmax(c, s) * x) * fmin(c, s);
	return (1.0 / t_0) / t_0;
}

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
    real(8) :: t_0
    t_0 = (fmax(c, s) * x) * fmin(c, s)
    code = (1.0d0 / t_0) / t_0
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (fmax(c, s) * x) * fmin(c, s);
	return (1.0 / t_0) / t_0;
}

Python

def code(x, c, s):
	t_0 = (fmax(c, s) * x) * fmin(c, s)
	return (1.0 / t_0) / t_0

Julia

function code(x, c, s)
	t_0 = Float64(Float64(fmax(c, s) * x) * fmin(c, s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (max(c, s) * x) * min(c, s);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(N[Max[c, s], $MachinePrecision] * x), $MachinePrecision] * N[Min[c, s], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

6. Applied rewrites 65.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-pow.f64 N/A

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

   16. pow2 N/A

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

   17. pow-prod-down N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. pow2 N/A

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

8. Applied rewrites 79.3%

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

Alternative 7: 79.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(c, s\right) \cdot \left(\mathsf{max}\left(c, s\right) \cdot x\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (fmin c s) (* (fmax c s) x)))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(c, s) * (fmax(c, s) * x);
	return 1.0 / (t_0 * t_0);
}

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
    real(8) :: t_0
    t_0 = fmin(c, s) * (fmax(c, s) * x)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(c, s) * (fmax(c, s) * x);
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = fmin(c, s) * (fmax(c, s) * x)
	return 1.0 / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(fmin(c, s) * Float64(fmax(c, s) * x))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = min(c, s) * (max(c, s) * x);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[Min[c, s], $MachinePrecision] * N[(N[Max[c, s], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

6. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 70.8%

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

8. Applied rewrites 70.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 79.1%

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 79.1%

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 79.1%

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

10. Applied rewrites 79.1%

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

Alternative 8: 76.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot x\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \mathbf{if}\;t\_0 \leq 8.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\left(t\_2 \cdot \left(t\_2 \cdot \left(t\_1 \cdot t\_0\right)\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(t\_2 \cdot x\right) \cdot \left(t\_2 \cdot t\_1\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s)))
        (t_1 (* t_0 x))
        (t_2 (fmin (fabs c) (fabs s))))
   (if (<= t_0 8.5e+168)
     (/ 1.0 (* (* t_2 (* t_2 (* t_1 t_0))) x))
     (/ 1.0 (* t_0 (* (* t_2 x) (* t_2 t_1)))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(fabs(c), fabs(s));
	double tmp;
	if (t_0 <= 8.5e+168) {
		tmp = 1.0 / ((t_2 * (t_2 * (t_1 * t_0))) * x);
	} else {
		tmp = 1.0 / (t_0 * ((t_2 * x) * (t_2 * t_1)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * x
    t_2 = fmin(abs(c), abs(s))
    if (t_0 <= 8.5d+168) then
        tmp = 1.0d0 / ((t_2 * (t_2 * (t_1 * t_0))) * x)
    else
        tmp = 1.0d0 / (t_0 * ((t_2 * x) * (t_2 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double tmp;
	if (t_0 <= 8.5e+168) {
		tmp = 1.0 / ((t_2 * (t_2 * (t_1 * t_0))) * x);
	} else {
		tmp = 1.0 / (t_0 * ((t_2 * x) * (t_2 * t_1)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * x
	t_2 = fmin(math.fabs(c), math.fabs(s))
	tmp = 0
	if t_0 <= 8.5e+168:
		tmp = 1.0 / ((t_2 * (t_2 * (t_1 * t_0))) * x)
	else:
		tmp = 1.0 / (t_0 * ((t_2 * x) * (t_2 * t_1)))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * x)
	t_2 = fmin(abs(c), abs(s))
	tmp = 0.0
	if (t_0 <= 8.5e+168)
		tmp = Float64(1.0 / Float64(Float64(t_2 * Float64(t_2 * Float64(t_1 * t_0))) * x));
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(t_2 * x) * Float64(t_2 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * x;
	t_2 = min(abs(c), abs(s));
	tmp = 0.0;
	if (t_0 <= 8.5e+168)
		tmp = 1.0 / ((t_2 * (t_2 * (t_1 * t_0))) * x);
	else
		tmp = 1.0 / (t_0 * ((t_2 * x) * (t_2 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 8.5e+168], N[(1.0 / N[(N[(t$95$2 * N[(t$95$2 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * N[(N[(t$95$2 * x), $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if s < 8.50000000000000069e168

   1. Initial program 66.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 70.8%

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

   8. Applied rewrites 70.8%

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

   if 8.50000000000000069e168 < s

   1. Initial program 66.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 70.8%

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

   8. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 76.0%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 76.0%

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

   10. Applied rewrites 76.0%

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

Alternative 9: 75.6% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* s (* (* c x) (* c (* s x))))))

C

double code(double x, double c, double s) {
	return 1.0 / (s * ((c * x) * (c * (s * 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 = 1.0d0 / (s * ((c * x) * (c * (s * x))))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (s * ((c * x) * (c * (s * x))));
}

Python

def code(x, c, s):
	return 1.0 / (s * ((c * x) * (c * (s * x))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(s * Float64(Float64(c * x) * Float64(c * Float64(s * x)))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (s * ((c * x) * (c * (s * x))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(s * N[(N[(c * x), $MachinePrecision] * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

6. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 70.8%

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

8. Applied rewrites 70.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 76.0%

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 76.0%

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

10. Applied rewrites 76.0%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))