mixedcos

Percentage Accurate: 66.5% → 97.9%

Time: 2.7s

Alternatives: 9

Speedup: 9.0×

• 68.7% 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 x)) (* (pow c 2) (* (* x (pow s 2)) 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 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2], $MachinePrecision] * N[(N[(x * N[Power[s, 2], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.5% 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 x)) (* (pow c 2) (* (* x (pow s 2)) 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 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2], $MachinePrecision] * N[(N[(x * N[Power[s, 2], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_0 \cdot t\_1\right) \cdot x\\ \mathbf{if}\;t\_0 \leq \frac{1668739871813211}{8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(t\_0 \cdot \left(\left(t\_0 \cdot x\right) \cdot t\_1\right)\right) \cdot x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t\_2}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (fmin (fabs c) (fabs s)))
       (t_2 (* (* t_0 t_1) x)))
  (if (<=
       t_0
       1668739871813211/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096)
    (/ (cos (* 2 x)) (* (* (* t_0 (* (* t_0 x) t_1)) x) t_1))
    (/ (/ (cos (+ x x)) t_2) t_2))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double t_2 = (t_0 * t_1) * x;
	double tmp;
	if (t_0 <= 2e-79) {
		tmp = cos((2.0 * x)) / (((t_0 * ((t_0 * x) * t_1)) * x) * t_1);
	} else {
		tmp = (cos((x + 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) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = (t_0 * t_1) * x
    if (t_0 <= 2d-79) then
        tmp = cos((2.0d0 * x)) / (((t_0 * ((t_0 * x) * t_1)) * x) * t_1)
    else
        tmp = (cos((x + 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 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	double t_2 = (t_0 * t_1) * x;
	double tmp;
	if (t_0 <= 2e-79) {
		tmp = Math.cos((2.0 * x)) / (((t_0 * ((t_0 * x) * t_1)) * x) * t_1);
	} else {
		tmp = (Math.cos((x + x)) / t_2) / t_2;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	t_2 = (t_0 * t_1) * x
	tmp = 0
	if t_0 <= 2e-79:
		tmp = math.cos((2.0 * x)) / (((t_0 * ((t_0 * x) * t_1)) * x) * t_1)
	else:
		tmp = (math.cos((x + x)) / t_2) / t_2
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	t_2 = Float64(Float64(t_0 * t_1) * x)
	tmp = 0.0
	if (t_0 <= 2e-79)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64(t_0 * Float64(Float64(t_0 * x) * t_1)) * x) * t_1));
	else
		tmp = Float64(Float64(cos(Float64(x + x)) / t_2) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	t_2 = (t_0 * t_1) * x;
	tmp = 0.0;
	if (t_0 <= 2e-79)
		tmp = cos((2.0 * x)) / (((t_0 * ((t_0 * x) * t_1)) * x) * t_1);
	else
		tmp = (cos((x + x)) / t_2) / t_2;
	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[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, 1668739871813211/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096], N[(N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(t$95$0 * N[(N[(t$95$0 * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if s < 2e-79

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.5%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 77.5%

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 77.5%

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

   3. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.4%

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

   5. Applied rewrites 91.4%

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

   if 2e-79 < s

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.4%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 78.3%

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 97.3%

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

Alternative 2: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \cos \left(\left|x\right| + \left|x\right|\right)\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := \left(t\_0 \cdot t\_2\right) \cdot \left|x\right|\\ t_4 := t\_2 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq \frac{7307508186654515}{2923003274661805836407369665432566039311865085952}:\\ \;\;\;\;\frac{\frac{1}{t\_2 \cdot \left(t\_0 \cdot \left|x\right|\right)}}{t\_3}\\ \mathbf{elif}\;\left|x\right| \leq 4999999999999999909315349154054740991463637108491892860888337397349569053269712469449300329851548412746772308261348178402514182220821421164656873275098572126930396830492460411478655642866237930786475017764864:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot \left(t\_4 \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\left|x\right| \cdot t\_2\right) \cdot \left(t\_0 \cdot \left(t\_0 \cdot t\_4\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (cos (+ (fabs x) (fabs x))))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* (* t_0 t_2) (fabs x)))
       (t_4 (* t_2 (fabs x))))
  (if (<=
       (fabs x)
       7307508186654515/2923003274661805836407369665432566039311865085952)
    (/ (/ 1 (* t_2 (* t_0 (fabs x)))) t_3)
    (if (<=
         (fabs x)
         4999999999999999909315349154054740991463637108491892860888337397349569053269712469449300329851548412746772308261348178402514182220821421164656873275098572126930396830492460411478655642866237930786475017764864)
      (/ t_1 (* t_0 (* t_4 t_3)))
      (/ t_1 (* (* (fabs x) t_2) (* t_0 (* t_0 t_4))))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = cos((fabs(x) + fabs(x)));
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = (t_0 * t_2) * fabs(x);
	double t_4 = t_2 * fabs(x);
	double tmp;
	if (fabs(x) <= 2.5e-33) {
		tmp = (1.0 / (t_2 * (t_0 * fabs(x)))) / t_3;
	} else if (fabs(x) <= 5e+207) {
		tmp = t_1 / (t_0 * (t_4 * t_3));
	} else {
		tmp = t_1 / ((fabs(x) * t_2) * (t_0 * (t_0 * t_4)));
	}
	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) :: t_4
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = cos((abs(x) + abs(x)))
    t_2 = fmin(abs(c), abs(s))
    t_3 = (t_0 * t_2) * abs(x)
    t_4 = t_2 * abs(x)
    if (abs(x) <= 2.5d-33) then
        tmp = (1.0d0 / (t_2 * (t_0 * abs(x)))) / t_3
    else if (abs(x) <= 5d+207) then
        tmp = t_1 / (t_0 * (t_4 * t_3))
    else
        tmp = t_1 / ((abs(x) * t_2) * (t_0 * (t_0 * t_4)))
    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 = Math.cos((Math.abs(x) + Math.abs(x)));
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = (t_0 * t_2) * Math.abs(x);
	double t_4 = t_2 * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 2.5e-33) {
		tmp = (1.0 / (t_2 * (t_0 * Math.abs(x)))) / t_3;
	} else if (Math.abs(x) <= 5e+207) {
		tmp = t_1 / (t_0 * (t_4 * t_3));
	} else {
		tmp = t_1 / ((Math.abs(x) * t_2) * (t_0 * (t_0 * t_4)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = math.cos((math.fabs(x) + math.fabs(x)))
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = (t_0 * t_2) * math.fabs(x)
	t_4 = t_2 * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 2.5e-33:
		tmp = (1.0 / (t_2 * (t_0 * math.fabs(x)))) / t_3
	elif math.fabs(x) <= 5e+207:
		tmp = t_1 / (t_0 * (t_4 * t_3))
	else:
		tmp = t_1 / ((math.fabs(x) * t_2) * (t_0 * (t_0 * t_4)))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = cos(Float64(abs(x) + abs(x)))
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(Float64(t_0 * t_2) * abs(x))
	t_4 = Float64(t_2 * abs(x))
	tmp = 0.0
	if (abs(x) <= 2.5e-33)
		tmp = Float64(Float64(1.0 / Float64(t_2 * Float64(t_0 * abs(x)))) / t_3);
	elseif (abs(x) <= 5e+207)
		tmp = Float64(t_1 / Float64(t_0 * Float64(t_4 * t_3)));
	else
		tmp = Float64(t_1 / Float64(Float64(abs(x) * t_2) * Float64(t_0 * Float64(t_0 * t_4))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = cos((abs(x) + abs(x)));
	t_2 = min(abs(c), abs(s));
	t_3 = (t_0 * t_2) * abs(x);
	t_4 = t_2 * abs(x);
	tmp = 0.0;
	if (abs(x) <= 2.5e-33)
		tmp = (1.0 / (t_2 * (t_0 * abs(x)))) / t_3;
	elseif (abs(x) <= 5e+207)
		tmp = t_1 / (t_0 * (t_4 * t_3));
	else
		tmp = t_1 / ((abs(x) * t_2) * (t_0 * (t_0 * t_4)));
	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[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 7307508186654515/2923003274661805836407369665432566039311865085952], N[(N[(1 / N[(t$95$2 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 4999999999999999909315349154054740991463637108491892860888337397349569053269712469449300329851548412746772308261348178402514182220821421164656873275098572126930396830492460411478655642866237930786475017764864], N[(t$95$1 / N[(t$95$0 * N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(N[Abs[x], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5000000000000001e-33

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.4%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 78.3%

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 77.4%

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

   10. Applied rewrites 77.4%

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

   if 2.5000000000000001e-33 < x < 4.9999999999999999e207

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.5%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 77.5%

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 77.5%

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

   3. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 90.2%

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

   5. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 90.2%

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

   7. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lower-*.f64 91.7%

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

   9. Applied rewrites 91.7%

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

   if 4.9999999999999999e207 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.2%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.2%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.2%

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

   3. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.9%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.9%

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

   5. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 92.9%

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

   7. Applied rewrites 92.9%

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

Alternative 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_0 \cdot t\_1\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq \frac{7307508186654515}{2923003274661805836407369665432566039311865085952}:\\ \;\;\;\;\frac{\frac{1}{t\_1 \cdot \left(t\_0 \cdot \left|x\right|\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_0 \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (fmin (fabs c) (fabs s)))
       (t_2 (* (* t_0 t_1) (fabs x))))
  (if (<=
       (fabs x)
       7307508186654515/2923003274661805836407369665432566039311865085952)
    (/ (/ 1 (* t_1 (* t_0 (fabs x)))) t_2)
    (/ (cos (+ (fabs x) (fabs x))) (* t_0 (* (* t_1 (fabs x)) t_2))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double t_2 = (t_0 * t_1) * fabs(x);
	double tmp;
	if (fabs(x) <= 2.5e-33) {
		tmp = (1.0 / (t_1 * (t_0 * fabs(x)))) / t_2;
	} else {
		tmp = cos((fabs(x) + fabs(x))) / (t_0 * ((t_1 * fabs(x)) * 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) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = (t_0 * t_1) * abs(x)
    if (abs(x) <= 2.5d-33) then
        tmp = (1.0d0 / (t_1 * (t_0 * abs(x)))) / t_2
    else
        tmp = cos((abs(x) + abs(x))) / (t_0 * ((t_1 * abs(x)) * t_2))
    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 = fmin(Math.abs(c), Math.abs(s));
	double t_2 = (t_0 * t_1) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 2.5e-33) {
		tmp = (1.0 / (t_1 * (t_0 * Math.abs(x)))) / t_2;
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / (t_0 * ((t_1 * Math.abs(x)) * t_2));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	t_2 = (t_0 * t_1) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 2.5e-33:
		tmp = (1.0 / (t_1 * (t_0 * math.fabs(x)))) / t_2
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / (t_0 * ((t_1 * math.fabs(x)) * t_2))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	t_2 = Float64(Float64(t_0 * t_1) * abs(x))
	tmp = 0.0
	if (abs(x) <= 2.5e-33)
		tmp = Float64(Float64(1.0 / Float64(t_1 * Float64(t_0 * abs(x)))) / t_2);
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(t_0 * Float64(Float64(t_1 * abs(x)) * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	t_2 = (t_0 * t_1) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 2.5e-33)
		tmp = (1.0 / (t_1 * (t_0 * abs(x)))) / t_2;
	else
		tmp = cos((abs(x) + abs(x))) / (t_0 * ((t_1 * abs(x)) * t_2));
	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[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 7307508186654515/2923003274661805836407369665432566039311865085952], N[(N[(1 / N[(t$95$1 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e-33

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 66.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.4%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.4%

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

   3. Applied rewrites 66.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 78.3%

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 77.4%

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

   10. Applied rewrites 77.4%

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

   if 2.5000000000000001e-33 < x

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.5%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 77.5%

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 77.5%

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

   3. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 90.2%

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

   5. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 90.2%

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

   7. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lower-*.f64 91.7%

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

   9. Applied rewrites 91.7%

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

Alternative 4: 78.4% accurate, 7.8× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (* (* s c) x))) (/ (/ 1 t_0) t_0)))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * 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 = (s * c) * x
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 78.2%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.2%

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 78.2%

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

3. Applied rewrites 78.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 66.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

8. Applied rewrites 78.4%

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

Alternative 5: 78.3% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (* (* s c) x))) (/ 1 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * 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 = (s * c) * x
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, N[(1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 78.2%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.2%

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 78.2%

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

3. Applied rewrites 78.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 78.6%

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

   22. lower-*.f64 77.3%

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 77.3%

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

   26. lift-*.f64 N/A

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

   27. lift-*.f64 N/A

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

   28. associate-*r* N/A

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

   29. *-commutative N/A

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

   30. lift-*.f64 N/A

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

8. Applied rewrites 78.3%

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

Alternative 6: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ \mathbf{if}\;t\_1 \leq 10000000000000000171775323872177191180393104084305455107732328445200031262781885420082626742861173182722545959543542834786931126445173006249634549465088:\\ \;\;\;\;\frac{1}{\left(x \cdot t\_0\right) \cdot \left(\left(\left(t\_1 \cdot t\_1\right) \cdot x\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t\_1 \cdot t\_0\right) \cdot \left(\left(\left(t\_1 \cdot x\right) \cdot x\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s))) (t_1 (fmax (fabs c) (fabs s))))
  (if (<=
       t_1
       10000000000000000171775323872177191180393104084305455107732328445200031262781885420082626742861173182722545959543542834786931126445173006249634549465088)
    (/ 1 (* (* x t_0) (* (* (* t_1 t_1) x) t_0)))
    (/ 1 (* (* t_1 t_0) (* (* (* t_1 x) x) t_0))))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double tmp;
	if (t_1 <= 1e+151) {
		tmp = 1.0 / ((x * t_0) * (((t_1 * t_1) * x) * t_0));
	} else {
		tmp = 1.0 / ((t_1 * t_0) * (((t_1 * x) * x) * t_0));
	}
	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) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    if (t_1 <= 1d+151) then
        tmp = 1.0d0 / ((x * t_0) * (((t_1 * t_1) * x) * t_0))
    else
        tmp = 1.0d0 / ((t_1 * t_0) * (((t_1 * x) * x) * t_0))
    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 = fmax(Math.abs(c), Math.abs(s));
	double tmp;
	if (t_1 <= 1e+151) {
		tmp = 1.0 / ((x * t_0) * (((t_1 * t_1) * x) * t_0));
	} else {
		tmp = 1.0 / ((t_1 * t_0) * (((t_1 * x) * x) * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	tmp = 0
	if t_1 <= 1e+151:
		tmp = 1.0 / ((x * t_0) * (((t_1 * t_1) * x) * t_0))
	else:
		tmp = 1.0 / ((t_1 * t_0) * (((t_1 * x) * x) * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	tmp = 0.0
	if (t_1 <= 1e+151)
		tmp = Float64(1.0 / Float64(Float64(x * t_0) * Float64(Float64(Float64(t_1 * t_1) * x) * t_0)));
	else
		tmp = Float64(1.0 / Float64(Float64(t_1 * t_0) * Float64(Float64(Float64(t_1 * x) * x) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	tmp = 0.0;
	if (t_1 <= 1e+151)
		tmp = 1.0 / ((x * t_0) * (((t_1 * t_1) * x) * t_0));
	else
		tmp = 1.0 / ((t_1 * t_0) * (((t_1 * x) * x) * t_0));
	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[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 10000000000000000171775323872177191180393104084305455107732328445200031262781885420082626742861173182722545959543542834786931126445173006249634549465088], N[(1 / N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[(N[(N[(t$95$1 * x), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < 1e151

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.2%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.2%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.2%

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

   3. Applied rewrites 78.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.8%

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

   if 1e151 < s

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.2%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.2%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.2%

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

   3. Applied rewrites 78.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 72.4%

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

   8. Applied rewrites 72.4%

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

Alternative 7: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(t\_0 \cdot t\_1\right) \cdot \left(\left(\left(t\_0 \cdot x\right) \cdot x\right) \cdot t\_1\right)} \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s))) (t_1 (fmin (fabs c) (fabs s))))
  (/ 1 (* (* t_0 t_1) (* (* (* t_0 x) x) t_1)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	return 1.0 / ((t_0 * t_1) * (((t_0 * x) * x) * t_1));
}

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
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    code = 1.0d0 / ((t_0 * t_1) * (((t_0 * x) * x) * t_1))
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 = fmin(Math.abs(c), Math.abs(s));
	return 1.0 / ((t_0 * t_1) * (((t_0 * x) * x) * t_1));
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	return 1.0 / ((t_0 * t_1) * (((t_0 * x) * x) * t_1))

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(t_0 * t_1) * Float64(Float64(Float64(t_0 * x) * x) * t_1)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	tmp = 1.0 / ((t_0 * t_1) * (((t_0 * x) * x) * t_1));
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[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1 / N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 78.2%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.2%

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 78.2%

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

3. Applied rewrites 78.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 72.4%

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

8. Applied rewrites 72.4%

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

Alternative 8: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(t\_0 \cdot t\_0\right) \cdot \left(\left(\left(t\_1 \cdot x\right) \cdot x\right) \cdot t\_1\right)} \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s))) (t_1 (fmax (fabs c) (fabs s))))
  (/ 1 (* (* t_0 t_0) (* (* (* t_1 x) x) t_1)))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * x) * t_1));
}

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
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    code = 1.0d0 / ((t_0 * t_0) * (((t_1 * x) * x) * t_1))
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 = fmax(Math.abs(c), Math.abs(s));
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * x) * t_1));
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * x) * t_1))

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(t_0 * t_0) * Float64(Float64(Float64(t_1 * x) * x) * t_1)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	tmp = 1.0 / ((t_0 * t_0) * (((t_1 * x) * x) * t_1));
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[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(N[(t$95$1 * x), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 78.2%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.2%

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 78.2%

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

3. Applied rewrites 78.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 58.6%

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

8. Applied rewrites 64.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 64.7%

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

10. Applied rewrites 64.7%

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

Alternative 9: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(t\_0 \cdot t\_0\right) \cdot \left(\left(\left(t\_1 \cdot x\right) \cdot t\_1\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s))) (t_1 (fmax (fabs c) (fabs s))))
  (/ 1 (* (* t_0 t_0) (* (* (* t_1 x) t_1) x)))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * t_1) * 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    code = 1.0d0 / ((t_0 * t_0) * (((t_1 * x) * t_1) * x))
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 = fmax(Math.abs(c), Math.abs(s));
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * t_1) * x));
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	return 1.0 / ((t_0 * t_0) * (((t_1 * x) * t_1) * x))

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(t_0 * t_0) * Float64(Float64(Float64(t_1 * x) * t_1) * x)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	tmp = 1.0 / ((t_0 * t_0) * (((t_1 * x) * t_1) * x));
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[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(N[(t$95$1 * x), $MachinePrecision] * t$95$1), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 78.2%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.2%

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 78.2%

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

3. Applied rewrites 78.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 58.6%

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

8. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2025274 -o generate:evaluate
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow c 2) (* (* x (pow s 2)) x))))