mixedcos

Percentage Accurate: 67.2% → 98.0%

Time: 1.2min

Alternatives: 14

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: 67.2% 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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := \cos \left(\left|x\right| + \left|x\right|\right)\\ t_4 := t\_2 \cdot t\_1\\ t_5 := t\_2 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq \frac{944473296573929}{18889465931478580854784}:\\ \;\;\;\;\frac{\frac{1}{t\_4} \cdot 1}{t\_4}\\ \mathbf{elif}\;\left|x\right| \leq 3799999999999999756058617039041391764068624108275936490937173340954074795328821049994414470557602482673561068656167289341512123668244335230203759534181088908978493199970248798074544917117736881042960550110507673365885163484577344450433556716453888:\\ \;\;\;\;\frac{1}{\left(\left(t\_5 \cdot \left|x\right|\right) \cdot \left(t\_2 \cdot t\_0\right)\right) \cdot t\_0} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\left(\left(t\_1 \cdot t\_0\right) \cdot t\_2\right) \cdot t\_5}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (cos (+ (fabs x) (fabs x))))
       (t_4 (* t_2 t_1))
       (t_5 (* t_2 (fabs x))))
  (if (<= (fabs x) 944473296573929/18889465931478580854784)
    (/ (* (/ 1 t_4) 1) t_4)
    (if (<=
         (fabs x)
         3799999999999999756058617039041391764068624108275936490937173340954074795328821049994414470557602482673561068656167289341512123668244335230203759534181088908978493199970248798074544917117736881042960550110507673365885163484577344450433556716453888)
      (* (/ 1 (* (* (* t_5 (fabs x)) (* t_2 t_0)) t_0)) t_3)
      (/ t_3 (* (* (* t_1 t_0) t_2) t_5))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * fabs(x);
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = cos((fabs(x) + fabs(x)));
	double t_4 = t_2 * t_1;
	double t_5 = t_2 * fabs(x);
	double tmp;
	if (fabs(x) <= 5e-8) {
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	} else if (fabs(x) <= 3.8e+246) {
		tmp = (1.0 / (((t_5 * fabs(x)) * (t_2 * t_0)) * t_0)) * t_3;
	} else {
		tmp = t_3 / (((t_1 * t_0) * t_2) * t_5);
	}
	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) :: t_5
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * abs(x)
    t_2 = fmin(abs(c), abs(s))
    t_3 = cos((abs(x) + abs(x)))
    t_4 = t_2 * t_1
    t_5 = t_2 * abs(x)
    if (abs(x) <= 5d-8) then
        tmp = ((1.0d0 / t_4) * 1.0d0) / t_4
    else if (abs(x) <= 3.8d+246) then
        tmp = (1.0d0 / (((t_5 * abs(x)) * (t_2 * t_0)) * t_0)) * t_3
    else
        tmp = t_3 / (((t_1 * t_0) * t_2) * t_5)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * Math.abs(x);
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = Math.cos((Math.abs(x) + Math.abs(x)));
	double t_4 = t_2 * t_1;
	double t_5 = t_2 * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 5e-8) {
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	} else if (Math.abs(x) <= 3.8e+246) {
		tmp = (1.0 / (((t_5 * Math.abs(x)) * (t_2 * t_0)) * t_0)) * t_3;
	} else {
		tmp = t_3 / (((t_1 * t_0) * t_2) * t_5);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * math.fabs(x)
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = math.cos((math.fabs(x) + math.fabs(x)))
	t_4 = t_2 * t_1
	t_5 = t_2 * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 5e-8:
		tmp = ((1.0 / t_4) * 1.0) / t_4
	elif math.fabs(x) <= 3.8e+246:
		tmp = (1.0 / (((t_5 * math.fabs(x)) * (t_2 * t_0)) * t_0)) * t_3
	else:
		tmp = t_3 / (((t_1 * t_0) * t_2) * t_5)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * abs(x))
	t_2 = fmin(abs(c), abs(s))
	t_3 = cos(Float64(abs(x) + abs(x)))
	t_4 = Float64(t_2 * t_1)
	t_5 = Float64(t_2 * abs(x))
	tmp = 0.0
	if (abs(x) <= 5e-8)
		tmp = Float64(Float64(Float64(1.0 / t_4) * 1.0) / t_4);
	elseif (abs(x) <= 3.8e+246)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(t_5 * abs(x)) * Float64(t_2 * t_0)) * t_0)) * t_3);
	else
		tmp = Float64(t_3 / Float64(Float64(Float64(t_1 * t_0) * t_2) * t_5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * abs(x);
	t_2 = min(abs(c), abs(s));
	t_3 = cos((abs(x) + abs(x)));
	t_4 = t_2 * t_1;
	t_5 = t_2 * abs(x);
	tmp = 0.0;
	if (abs(x) <= 5e-8)
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	elseif (abs(x) <= 3.8e+246)
		tmp = (1.0 / (((t_5 * abs(x)) * (t_2 * t_0)) * t_0)) * t_3;
	else
		tmp = t_3 / (((t_1 * t_0) * t_2) * t_5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 944473296573929/18889465931478580854784], N[(N[(N[(1 / t$95$4), $MachinePrecision] * 1), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 3799999999999999756058617039041391764068624108275936490937173340954074795328821049994414470557602482673561068656167289341512123668244335230203759534181088908978493199970248798074544917117736881042960550110507673365885163484577344450433556716453888], N[(N[(1 / N[(N[(N[(t$95$5 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], N[(t$95$3 / N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999998e-8

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 4.9999999999999998e-8 < x < 3.7999999999999998e246

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-/r* N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. mult-flip N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 85.7%

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

   7. Applied rewrites 85.7%

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

   if 3.7999999999999998e246 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 79.0%

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

   5. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.7%

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

   7. Applied rewrites 85.7%

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

Alternative 2: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \cos \left(\left|x\right| + \left|x\right|\right)\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot \left(t\_0 \cdot \left|x\right|\right)\\ t_4 := t\_2 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq \frac{237684487542793}{2475880078570760549798248448}:\\ \;\;\;\;\frac{\frac{1}{t\_3} \cdot 1}{t\_3}\\ \mathbf{elif}\;\left|x\right| \leq 50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368:\\ \;\;\;\;\frac{1}{\left(t\_2 \cdot \left(\left(t\_4 \cdot \left|x\right|\right) \cdot t\_0\right)\right) \cdot t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(t\_4 \cdot t\_4\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \]

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_2 (* t_0 (fabs x))))
       (t_4 (* t_2 (fabs x))))
  (if (<= (fabs x) 237684487542793/2475880078570760549798248448)
    (/ (* (/ 1 t_3) 1) t_3)
    (if (<=
         (fabs x)
         50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368)
      (* (/ 1 (* (* t_2 (* (* t_4 (fabs x)) t_0)) t_0)) t_1)
      (/ t_1 (* (* t_4 t_4) (* t_0 t_0)))))))

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_2 * (t_0 * fabs(x));
	double t_4 = t_2 * fabs(x);
	double tmp;
	if (fabs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else if (fabs(x) <= 5e+181) {
		tmp = (1.0 / ((t_2 * ((t_4 * fabs(x)) * t_0)) * t_0)) * t_1;
	} else {
		tmp = t_1 / ((t_4 * t_4) * (t_0 * 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) :: 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_2 * (t_0 * abs(x))
    t_4 = t_2 * abs(x)
    if (abs(x) <= 9.6d-14) then
        tmp = ((1.0d0 / t_3) * 1.0d0) / t_3
    else if (abs(x) <= 5d+181) then
        tmp = (1.0d0 / ((t_2 * ((t_4 * abs(x)) * t_0)) * t_0)) * t_1
    else
        tmp = t_1 / ((t_4 * t_4) * (t_0 * t_0))
    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_2 * (t_0 * Math.abs(x));
	double t_4 = t_2 * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else if (Math.abs(x) <= 5e+181) {
		tmp = (1.0 / ((t_2 * ((t_4 * Math.abs(x)) * t_0)) * t_0)) * t_1;
	} else {
		tmp = t_1 / ((t_4 * t_4) * (t_0 * t_0));
	}
	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_2 * (t_0 * math.fabs(x))
	t_4 = t_2 * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 9.6e-14:
		tmp = ((1.0 / t_3) * 1.0) / t_3
	elif math.fabs(x) <= 5e+181:
		tmp = (1.0 / ((t_2 * ((t_4 * math.fabs(x)) * t_0)) * t_0)) * t_1
	else:
		tmp = t_1 / ((t_4 * t_4) * (t_0 * t_0))
	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(t_2 * Float64(t_0 * abs(x)))
	t_4 = Float64(t_2 * abs(x))
	tmp = 0.0
	if (abs(x) <= 9.6e-14)
		tmp = Float64(Float64(Float64(1.0 / t_3) * 1.0) / t_3);
	elseif (abs(x) <= 5e+181)
		tmp = Float64(Float64(1.0 / Float64(Float64(t_2 * Float64(Float64(t_4 * abs(x)) * t_0)) * t_0)) * t_1);
	else
		tmp = Float64(t_1 / Float64(Float64(t_4 * t_4) * Float64(t_0 * t_0)));
	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_2 * (t_0 * abs(x));
	t_4 = t_2 * abs(x);
	tmp = 0.0;
	if (abs(x) <= 9.6e-14)
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	elseif (abs(x) <= 5e+181)
		tmp = (1.0 / ((t_2 * ((t_4 * abs(x)) * t_0)) * t_0)) * t_1;
	else
		tmp = t_1 / ((t_4 * t_4) * (t_0 * t_0));
	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[(t$95$2 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 237684487542793/2475880078570760549798248448], N[(N[(N[(1 / t$95$3), $MachinePrecision] * 1), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368], N[(N[(1 / N[(N[(t$95$2 * N[(N[(t$95$4 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 / N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5999999999999995e-14

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 9.5999999999999995e-14 < x < 5.0000000000000003e181

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-/r* N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. mult-flip N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.7%

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

   7. Applied rewrites 85.7%

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

   if 5.0000000000000003e181 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 77.6%

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

   5. Applied rewrites 77.6%

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

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot t\_1\\ t_4 := t\_2 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq \frac{237684487542793}{2475880078570760549798248448}:\\ \;\;\;\;\frac{\frac{1}{t\_3} \cdot 1}{t\_3}\\ \mathbf{elif}\;\left|x\right| \leq 50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368:\\ \;\;\;\;\frac{\cos \left(2 \cdot \left|x\right|\right)}{\left(t\_2 \cdot \left(t\_4 \cdot t\_1\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_4 \cdot t\_4\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* t_2 t_1))
       (t_4 (* t_2 (fabs x))))
  (if (<= (fabs x) 237684487542793/2475880078570760549798248448)
    (/ (* (/ 1 t_3) 1) t_3)
    (if (<=
         (fabs x)
         50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368)
      (/ (cos (* 2 (fabs x))) (* (* t_2 (* t_4 t_1)) t_0))
      (/ (cos (+ (fabs x) (fabs x))) (* (* t_4 t_4) (* t_0 t_0)))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * fabs(x);
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * t_1;
	double t_4 = t_2 * fabs(x);
	double tmp;
	if (fabs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else if (fabs(x) <= 5e+181) {
		tmp = cos((2.0 * fabs(x))) / ((t_2 * (t_4 * t_1)) * t_0);
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((t_4 * t_4) * (t_0 * 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * abs(x)
    t_2 = fmin(abs(c), abs(s))
    t_3 = t_2 * t_1
    t_4 = t_2 * abs(x)
    if (abs(x) <= 9.6d-14) then
        tmp = ((1.0d0 / t_3) * 1.0d0) / t_3
    else if (abs(x) <= 5d+181) then
        tmp = cos((2.0d0 * abs(x))) / ((t_2 * (t_4 * t_1)) * t_0)
    else
        tmp = cos((abs(x) + abs(x))) / ((t_4 * t_4) * (t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * Math.abs(x);
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = t_2 * t_1;
	double t_4 = t_2 * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else if (Math.abs(x) <= 5e+181) {
		tmp = Math.cos((2.0 * Math.abs(x))) / ((t_2 * (t_4 * t_1)) * t_0);
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_4 * t_4) * (t_0 * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * math.fabs(x)
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = t_2 * t_1
	t_4 = t_2 * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 9.6e-14:
		tmp = ((1.0 / t_3) * 1.0) / t_3
	elif math.fabs(x) <= 5e+181:
		tmp = math.cos((2.0 * math.fabs(x))) / ((t_2 * (t_4 * t_1)) * t_0)
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_4 * t_4) * (t_0 * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * abs(x))
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * t_1)
	t_4 = Float64(t_2 * abs(x))
	tmp = 0.0
	if (abs(x) <= 9.6e-14)
		tmp = Float64(Float64(Float64(1.0 / t_3) * 1.0) / t_3);
	elseif (abs(x) <= 5e+181)
		tmp = Float64(cos(Float64(2.0 * abs(x))) / Float64(Float64(t_2 * Float64(t_4 * t_1)) * t_0));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_4 * t_4) * Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * abs(x);
	t_2 = min(abs(c), abs(s));
	t_3 = t_2 * t_1;
	t_4 = t_2 * abs(x);
	tmp = 0.0;
	if (abs(x) <= 9.6e-14)
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	elseif (abs(x) <= 5e+181)
		tmp = cos((2.0 * abs(x))) / ((t_2 * (t_4 * t_1)) * t_0);
	else
		tmp = cos((abs(x) + abs(x))) / ((t_4 * t_4) * (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 237684487542793/2475880078570760549798248448], N[(N[(N[(1 / t$95$3), $MachinePrecision] * 1), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 50000000000000003226559936361919779827105376205144584884917978916367904662510143278135754996687257850822691394447592090096097397546144525317852447661395664561828975608881910401466368], N[(N[Cos[N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$2 * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5999999999999995e-14

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 9.5999999999999995e-14 < x < 5.0000000000000003e181

   1. Initial program 67.2%

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

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.7%

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

   3. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.0%

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

   5. Applied rewrites 90.0%

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

   if 5.0000000000000003e181 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 77.6%

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

   5. Applied rewrites 77.6%

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

Alternative 4: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \cos \left(\left|x\right| + \left|x\right|\right)\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot \left|x\right|\\ t_4 := t\_2 \cdot \left(t\_0 \cdot \left|x\right|\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{237684487542793}{2475880078570760549798248448}:\\ \;\;\;\;\frac{\frac{1}{t\_4} \cdot 1}{t\_4}\\ \mathbf{elif}\;\left|x\right| \leq 500000000000000034478783768422291468839913049176218549546891415298328160321104377283093399808452642713299991464708729440150191950239130597851790859288683698879916192875675648:\\ \;\;\;\;\frac{t\_1}{\left(\left(t\_4 \cdot \left|x\right|\right) \cdot t\_2\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(t\_3 \cdot t\_3\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \]

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_2 (fabs x)))
       (t_4 (* t_2 (* t_0 (fabs x)))))
  (if (<= (fabs x) 237684487542793/2475880078570760549798248448)
    (/ (* (/ 1 t_4) 1) t_4)
    (if (<=
         (fabs x)
         500000000000000034478783768422291468839913049176218549546891415298328160321104377283093399808452642713299991464708729440150191950239130597851790859288683698879916192875675648)
      (/ t_1 (* (* (* t_4 (fabs x)) t_2) t_0))
      (/ t_1 (* (* t_3 t_3) (* t_0 t_0)))))))

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_2 * fabs(x);
	double t_4 = t_2 * (t_0 * fabs(x));
	double tmp;
	if (fabs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	} else if (fabs(x) <= 5e+173) {
		tmp = t_1 / (((t_4 * fabs(x)) * t_2) * t_0);
	} else {
		tmp = t_1 / ((t_3 * t_3) * (t_0 * 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) :: 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_2 * abs(x)
    t_4 = t_2 * (t_0 * abs(x))
    if (abs(x) <= 9.6d-14) then
        tmp = ((1.0d0 / t_4) * 1.0d0) / t_4
    else if (abs(x) <= 5d+173) then
        tmp = t_1 / (((t_4 * abs(x)) * t_2) * t_0)
    else
        tmp = t_1 / ((t_3 * t_3) * (t_0 * t_0))
    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_2 * Math.abs(x);
	double t_4 = t_2 * (t_0 * Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	} else if (Math.abs(x) <= 5e+173) {
		tmp = t_1 / (((t_4 * Math.abs(x)) * t_2) * t_0);
	} else {
		tmp = t_1 / ((t_3 * t_3) * (t_0 * t_0));
	}
	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_2 * math.fabs(x)
	t_4 = t_2 * (t_0 * math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 9.6e-14:
		tmp = ((1.0 / t_4) * 1.0) / t_4
	elif math.fabs(x) <= 5e+173:
		tmp = t_1 / (((t_4 * math.fabs(x)) * t_2) * t_0)
	else:
		tmp = t_1 / ((t_3 * t_3) * (t_0 * t_0))
	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(t_2 * abs(x))
	t_4 = Float64(t_2 * Float64(t_0 * abs(x)))
	tmp = 0.0
	if (abs(x) <= 9.6e-14)
		tmp = Float64(Float64(Float64(1.0 / t_4) * 1.0) / t_4);
	elseif (abs(x) <= 5e+173)
		tmp = Float64(t_1 / Float64(Float64(Float64(t_4 * abs(x)) * t_2) * t_0));
	else
		tmp = Float64(t_1 / Float64(Float64(t_3 * t_3) * Float64(t_0 * t_0)));
	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_2 * abs(x);
	t_4 = t_2 * (t_0 * abs(x));
	tmp = 0.0;
	if (abs(x) <= 9.6e-14)
		tmp = ((1.0 / t_4) * 1.0) / t_4;
	elseif (abs(x) <= 5e+173)
		tmp = t_1 / (((t_4 * abs(x)) * t_2) * t_0);
	else
		tmp = t_1 / ((t_3 * t_3) * (t_0 * t_0));
	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[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 237684487542793/2475880078570760549798248448], N[(N[(N[(1 / t$95$4), $MachinePrecision] * 1), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 500000000000000034478783768422291468839913049176218549546891415298328160321104377283093399808452642713299991464708729440150191950239130597851790859288683698879916192875675648], N[(t$95$1 / N[(N[(N[(t$95$4 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t$95$3 * t$95$3), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5999999999999995e-14

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 9.5999999999999995e-14 < x < 5.0000000000000003e173

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 78.3%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 78.3%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 78.3%

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

   3. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.8%

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

   5. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 91.6%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.6%

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 90.0%

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

   if 5.0000000000000003e173 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 77.6%

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

   5. Applied rewrites 77.6%

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

Alternative 5: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot t\_1\\ \mathbf{if}\;\left|x\right| \leq \frac{237684487542793}{2475880078570760549798248448}:\\ \;\;\;\;\frac{\frac{1}{t\_3} \cdot 1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot \left|x\right|\right)}{\left(\left(t\_1 \cdot t\_2\right) \cdot \left(t\_2 \cdot \left|x\right|\right)\right) \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* t_2 t_1)))
  (if (<= (fabs x) 237684487542793/2475880078570760549798248448)
    (/ (* (/ 1 t_3) 1) t_3)
    (/
     (cos (* 2 (fabs x)))
     (* (* (* t_1 t_2) (* t_2 (fabs x))) t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * fabs(x);
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if (fabs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else {
		tmp = cos((2.0 * fabs(x))) / (((t_1 * t_2) * (t_2 * fabs(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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * abs(x)
    t_2 = fmin(abs(c), abs(s))
    t_3 = t_2 * t_1
    if (abs(x) <= 9.6d-14) then
        tmp = ((1.0d0 / t_3) * 1.0d0) / t_3
    else
        tmp = cos((2.0d0 * abs(x))) / (((t_1 * t_2) * (t_2 * abs(x))) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * Math.abs(x);
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if (Math.abs(x) <= 9.6e-14) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else {
		tmp = Math.cos((2.0 * Math.abs(x))) / (((t_1 * t_2) * (t_2 * Math.abs(x))) * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * math.fabs(x)
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = t_2 * t_1
	tmp = 0
	if math.fabs(x) <= 9.6e-14:
		tmp = ((1.0 / t_3) * 1.0) / t_3
	else:
		tmp = math.cos((2.0 * math.fabs(x))) / (((t_1 * t_2) * (t_2 * math.fabs(x))) * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * abs(x))
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * t_1)
	tmp = 0.0
	if (abs(x) <= 9.6e-14)
		tmp = Float64(Float64(Float64(1.0 / t_3) * 1.0) / t_3);
	else
		tmp = Float64(cos(Float64(2.0 * abs(x))) / Float64(Float64(Float64(t_1 * t_2) * Float64(t_2 * abs(x))) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * abs(x);
	t_2 = min(abs(c), abs(s));
	t_3 = t_2 * t_1;
	tmp = 0.0;
	if (abs(x) <= 9.6e-14)
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	else
		tmp = cos((2.0 * abs(x))) / (((t_1 * t_2) * (t_2 * abs(x))) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 237684487542793/2475880078570760549798248448], N[(N[(N[(1 / t$95$3), $MachinePrecision] * 1), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Cos[N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5999999999999995e-14

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 9.5999999999999995e-14 < x

   1. Initial program 67.2%

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

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.7%

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

   3. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 91.3%

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

   5. Applied rewrites 91.3%

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

Alternative 6: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_1 \cdot \left|x\right|\\ t_3 := t\_1 \cdot \left(t\_0 \cdot \left|x\right|\right)\\ \mathbf{if}\;\left|x\right| \leq 15999999999999999731564544:\\ \;\;\;\;\frac{\frac{1}{t\_3} \cdot 1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_2 \cdot t\_2\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \]

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_1 (fabs x)))
       (t_3 (* t_1 (* t_0 (fabs x)))))
  (if (<= (fabs x) 15999999999999999731564544)
    (/ (* (/ 1 t_3) 1) t_3)
    (/ (cos (+ (fabs x) (fabs x))) (* (* t_2 t_2) (* t_0 t_0))))))

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_1 * fabs(x);
	double t_3 = t_1 * (t_0 * fabs(x));
	double tmp;
	if (fabs(x) <= 1.6e+25) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((t_2 * t_2) * (t_0 * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = t_1 * abs(x)
    t_3 = t_1 * (t_0 * abs(x))
    if (abs(x) <= 1.6d+25) then
        tmp = ((1.0d0 / t_3) * 1.0d0) / t_3
    else
        tmp = cos((abs(x) + abs(x))) / ((t_2 * t_2) * (t_0 * t_0))
    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_1 * Math.abs(x);
	double t_3 = t_1 * (t_0 * Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 1.6e+25) {
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_2 * t_2) * (t_0 * t_0));
	}
	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_1 * math.fabs(x)
	t_3 = t_1 * (t_0 * math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 1.6e+25:
		tmp = ((1.0 / t_3) * 1.0) / t_3
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_2 * t_2) * (t_0 * t_0))
	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(t_1 * abs(x))
	t_3 = Float64(t_1 * Float64(t_0 * abs(x)))
	tmp = 0.0
	if (abs(x) <= 1.6e+25)
		tmp = Float64(Float64(Float64(1.0 / t_3) * 1.0) / t_3);
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_2 * t_2) * Float64(t_0 * t_0)));
	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_1 * abs(x);
	t_3 = t_1 * (t_0 * abs(x));
	tmp = 0.0;
	if (abs(x) <= 1.6e+25)
		tmp = ((1.0 / t_3) * 1.0) / t_3;
	else
		tmp = cos((abs(x) + abs(x))) / ((t_2 * t_2) * (t_0 * t_0));
	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[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 15999999999999999731564544], N[(N[(N[(1 / t$95$3), $MachinePrecision] * 1), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e25

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 1.6e25 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 77.6%

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

   5. Applied rewrites 77.6%

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

Alternative 7: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_1 \cdot \left(t\_0 \cdot \left|x\right|\right)\\ \mathbf{if}\;\left|x\right| \leq 15999999999999999731564544:\\ \;\;\;\;\frac{\frac{1}{t\_2} \cdot 1}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_1 \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \]

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_1 (* t_0 (fabs x)))))
  (if (<= (fabs x) 15999999999999999731564544)
    (/ (* (/ 1 t_2) 1) t_2)
    (/
     (cos (+ (fabs x) (fabs x)))
     (* (* t_1 (* (* t_1 (fabs x)) (fabs x))) (* t_0 t_0))))))

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_1 * (t_0 * fabs(x));
	double tmp;
	if (fabs(x) <= 1.6e+25) {
		tmp = ((1.0 / t_2) * 1.0) / t_2;
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((t_1 * ((t_1 * fabs(x)) * fabs(x))) * (t_0 * 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) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = t_1 * (t_0 * abs(x))
    if (abs(x) <= 1.6d+25) then
        tmp = ((1.0d0 / t_2) * 1.0d0) / t_2
    else
        tmp = cos((abs(x) + abs(x))) / ((t_1 * ((t_1 * abs(x)) * abs(x))) * (t_0 * t_0))
    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_1 * (t_0 * Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 1.6e+25) {
		tmp = ((1.0 / t_2) * 1.0) / t_2;
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_1 * ((t_1 * Math.abs(x)) * Math.abs(x))) * (t_0 * t_0));
	}
	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_1 * (t_0 * math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 1.6e+25:
		tmp = ((1.0 / t_2) * 1.0) / t_2
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_1 * ((t_1 * math.fabs(x)) * math.fabs(x))) * (t_0 * t_0))
	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(t_1 * Float64(t_0 * abs(x)))
	tmp = 0.0
	if (abs(x) <= 1.6e+25)
		tmp = Float64(Float64(Float64(1.0 / t_2) * 1.0) / t_2);
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_1 * Float64(Float64(t_1 * abs(x)) * abs(x))) * Float64(t_0 * t_0)));
	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_1 * (t_0 * abs(x));
	tmp = 0.0;
	if (abs(x) <= 1.6e+25)
		tmp = ((1.0 / t_2) * 1.0) / t_2;
	else
		tmp = cos((abs(x) + abs(x))) / ((t_1 * ((t_1 * abs(x)) * abs(x))) * (t_0 * t_0));
	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[(t$95$1 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 15999999999999999731564544], N[(N[(N[(1 / t$95$2), $MachinePrecision] * 1), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$1 * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e25

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

   10. Applied rewrites 79.0%

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

   if 1.6e25 < x

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

Alternative 8: 80.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 67.2%

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

   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 67.2%

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

   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 67.2%

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

   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 67.2%

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

3. Applied rewrites 67.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 59.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

8. Applied rewrites 76.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. mult-flip N/A

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

10. Applied rewrites 79.0%

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

Alternative 9: 79.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|c\right|\right) \cdot x\right) \cdot s\\ t_1 := \left|c\right| \cdot s\\ \mathbf{if}\;\left|c\right| \leq \frac{1162941958872971}{11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496}:\\ \;\;\;\;\frac{1}{t\_1 \cdot \left(\left(t\_1 \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (* (* (- (fabs c)) x) s)) (t_1 (* (fabs c) s)))
  (if (<=
       (fabs c)
       1162941958872971/11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496)
    (/ 1 (* t_1 (* (* t_1 x) x)))
    (/ 1 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (-fabs(c) * x) * s;
	double t_1 = fabs(c) * s;
	double tmp;
	if (fabs(c) <= 1e-121) {
		tmp = 1.0 / (t_1 * ((t_1 * x) * x));
	} else {
		tmp = 1.0 / (t_0 * 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 = (-abs(c) * x) * s
    t_1 = abs(c) * s
    if (abs(c) <= 1d-121) then
        tmp = 1.0d0 / (t_1 * ((t_1 * x) * x))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (-Math.abs(c) * x) * s;
	double t_1 = Math.abs(c) * s;
	double tmp;
	if (Math.abs(c) <= 1e-121) {
		tmp = 1.0 / (t_1 * ((t_1 * x) * x));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (-math.fabs(c) * x) * s
	t_1 = math.fabs(c) * s
	tmp = 0
	if math.fabs(c) <= 1e-121:
		tmp = 1.0 / (t_1 * ((t_1 * x) * x))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(Float64(-abs(c)) * x) * s)
	t_1 = Float64(abs(c) * s)
	tmp = 0.0
	if (abs(c) <= 1e-121)
		tmp = Float64(1.0 / Float64(t_1 * Float64(Float64(t_1 * x) * x)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (-abs(c) * x) * s;
	t_1 = abs(c) * s;
	tmp = 0.0;
	if (abs(c) <= 1e-121)
		tmp = 1.0 / (t_1 * ((t_1 * x) * x));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[((-N[Abs[c], $MachinePrecision]) * x), $MachinePrecision] * s), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[c], $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[Abs[c], $MachinePrecision], 1162941958872971/11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496], N[(1 / N[(t$95$1 * N[(N[(t$95$1 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999998e-122

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 76.8%

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

   10. Applied rewrites 76.8%

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

   if 9.9999999999999998e-122 < c

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. unswap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

   8. Applied rewrites 78.9%

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

Alternative 10: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot x\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \mathbf{if}\;t\_2 \leq \frac{4253529586511731}{42535295865117307932921825928971026432}:\\ \;\;\;\;\frac{1}{\left(\left(t\_1 \cdot \left(t\_2 \cdot t\_0\right)\right) \cdot t\_2\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(t\_2 \cdot t\_2\right) \cdot x\right) \cdot t\_0\right) \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* t_0 x))
       (t_2 (fmin (fabs c) (fabs s))))
  (if (<= t_2 4253529586511731/42535295865117307932921825928971026432)
    (/ 1 (* (* (* t_1 (* t_2 t_0)) t_2) x))
    (/ 1 (* (* (* (* t_2 t_2) x) t_0) t_1)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(fabs(c), fabs(s));
	double tmp;
	if (t_2 <= 1e-22) {
		tmp = 1.0 / (((t_1 * (t_2 * t_0)) * t_2) * x);
	} else {
		tmp = 1.0 / ((((t_2 * t_2) * x) * t_0) * t_1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * x
    t_2 = fmin(abs(c), abs(s))
    if (t_2 <= 1d-22) then
        tmp = 1.0d0 / (((t_1 * (t_2 * t_0)) * t_2) * x)
    else
        tmp = 1.0d0 / ((((t_2 * t_2) * x) * t_0) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double tmp;
	if (t_2 <= 1e-22) {
		tmp = 1.0 / (((t_1 * (t_2 * t_0)) * t_2) * x);
	} else {
		tmp = 1.0 / ((((t_2 * t_2) * x) * t_0) * t_1);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * x
	t_2 = fmin(math.fabs(c), math.fabs(s))
	tmp = 0
	if t_2 <= 1e-22:
		tmp = 1.0 / (((t_1 * (t_2 * t_0)) * t_2) * x)
	else:
		tmp = 1.0 / ((((t_2 * t_2) * x) * t_0) * t_1)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * x)
	t_2 = fmin(abs(c), abs(s))
	tmp = 0.0
	if (t_2 <= 1e-22)
		tmp = Float64(1.0 / Float64(Float64(Float64(t_1 * Float64(t_2 * t_0)) * t_2) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(t_2 * t_2) * x) * t_0) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * x;
	t_2 = min(abs(c), abs(s));
	tmp = 0.0;
	if (t_2 <= 1e-22)
		tmp = 1.0 / (((t_1 * (t_2 * t_0)) * t_2) * x);
	else
		tmp = 1.0 / ((((t_2 * t_2) * x) * t_0) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4253529586511731/42535295865117307932921825928971026432], N[(1 / N[(N[(N[(t$95$1 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1e-22

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 75.4%

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

   10. Applied rewrites 75.4%

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

   if 1e-22 < c

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.4%

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

   8. Applied rewrites 67.4%

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

Alternative 11: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_0 \leq \frac{3794275180128377}{948568795032094272909893509191171341133987714380927500611236528192824358010355712}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(\left(t\_2 \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(t\_0 \cdot t\_0\right) \cdot x\right) \cdot t\_1\right) \cdot \left(t\_1 \cdot x\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* t_0 t_1)))
  (if (<=
       t_0
       3794275180128377/948568795032094272909893509191171341133987714380927500611236528192824358010355712)
    (/ 1 (* t_2 (* (* t_2 x) x)))
    (/ 1 (* (* (* (* t_0 t_0) x) t_1) (* 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));
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_0 <= 4e-66) {
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	} else {
		tmp = 1.0 / ((((t_0 * t_0) * x) * t_1) * (t_1 * x));
	}
	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 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = t_0 * t_1
    if (t_0 <= 4d-66) then
        tmp = 1.0d0 / (t_2 * ((t_2 * x) * x))
    else
        tmp = 1.0d0 / ((((t_0 * t_0) * x) * t_1) * (t_1 * x))
    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 t_2 = t_0 * t_1;
	double tmp;
	if (t_0 <= 4e-66) {
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	} else {
		tmp = 1.0 / ((((t_0 * t_0) * x) * t_1) * (t_1 * x));
	}
	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))
	t_2 = t_0 * t_1
	tmp = 0
	if t_0 <= 4e-66:
		tmp = 1.0 / (t_2 * ((t_2 * x) * x))
	else:
		tmp = 1.0 / ((((t_0 * t_0) * x) * t_1) * (t_1 * x))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_0 <= 4e-66)
		tmp = Float64(1.0 / Float64(t_2 * Float64(Float64(t_2 * x) * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(t_0 * t_0) * x) * t_1) * Float64(t_1 * x)));
	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));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (t_0 <= 4e-66)
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	else
		tmp = 1.0 / ((((t_0 * t_0) * x) * t_1) * (t_1 * x));
	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]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, 3794275180128377/948568795032094272909893509191171341133987714380927500611236528192824358010355712], N[(1 / N[(t$95$2 * N[(N[(t$95$2 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 3.9999999999999999e-66

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 76.8%

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

   10. Applied rewrites 76.8%

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

   if 3.9999999999999999e-66 < c

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.4%

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

   8. Applied rewrites 67.4%

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

Alternative 12: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_0 \leq \frac{1162941958872971}{11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(\left(t\_2 \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t\_2 \cdot \left(t\_0 \cdot x\right)\right) \cdot \left(t\_1 \cdot x\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* t_0 t_1)))
  (if (<=
       t_0
       1162941958872971/11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496)
    (/ 1 (* t_2 (* (* t_2 x) x)))
    (/ 1 (* (* t_2 (* t_0 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));
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_0 <= 1e-121) {
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	} else {
		tmp = 1.0 / ((t_2 * (t_0 * x)) * (t_1 * x));
	}
	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 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = t_0 * t_1
    if (t_0 <= 1d-121) then
        tmp = 1.0d0 / (t_2 * ((t_2 * x) * x))
    else
        tmp = 1.0d0 / ((t_2 * (t_0 * x)) * (t_1 * x))
    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 t_2 = t_0 * t_1;
	double tmp;
	if (t_0 <= 1e-121) {
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	} else {
		tmp = 1.0 / ((t_2 * (t_0 * x)) * (t_1 * x));
	}
	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))
	t_2 = t_0 * t_1
	tmp = 0
	if t_0 <= 1e-121:
		tmp = 1.0 / (t_2 * ((t_2 * x) * x))
	else:
		tmp = 1.0 / ((t_2 * (t_0 * x)) * (t_1 * x))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_0 <= 1e-121)
		tmp = Float64(1.0 / Float64(t_2 * Float64(Float64(t_2 * x) * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(t_2 * Float64(t_0 * x)) * Float64(t_1 * x)));
	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));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (t_0 <= 1e-121)
		tmp = 1.0 / (t_2 * ((t_2 * x) * x));
	else
		tmp = 1.0 / ((t_2 * (t_0 * x)) * (t_1 * x));
	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]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, 1162941958872971/11629419588729710248789180926208072549658261770997088964503843186890228609814366773219056811420217048972200345700258846936553626057834496], N[(1 / N[(t$95$2 * N[(N[(t$95$2 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(t$95$2 * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999998e-122

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 76.8%

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

   10. Applied rewrites 76.8%

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

   if 9.9999999999999998e-122 < c

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.2%

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

      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 67.2%

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

      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 67.2%

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

      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 67.2%

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

   3. Applied rewrites 67.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.4%

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

   10. Applied rewrites 75.4%

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

Alternative 13: 77.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s))) (t_1 (fmax (fabs c) (fabs s))))
  (/ 1 (* (* (* (* (* t_1 x) t_0) t_1) 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));
	return 1.0 / (((((t_1 * x) * t_0) * t_1) * x) * 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
    real(8) :: t_1
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    code = 1.0d0 / (((((t_1 * x) * t_0) * t_1) * x) * t_0)
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_1 * x) * t_0) * t_1) * x) * t_0);
}

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_1 * x) * t_0) * t_1) * x) * t_0)

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(Float64(Float64(Float64(t_1 * x) * t_0) * t_1) * x) * t_0))
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_1 * x) * t_0) * t_1) * x) * t_0);
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[(N[(N[(N[(t$95$1 * x), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 67.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 67.2%

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

   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 67.2%

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

   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 67.2%

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

   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 67.2%

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

3. Applied rewrites 67.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 59.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

8. Applied rewrites 76.3%

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

Alternative 14: 76.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * s) * (((c * s) * x) * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(1 / N[(N[(c * s), $MachinePrecision] * N[(N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 67.2%

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

   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 67.2%

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

   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 67.2%

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

   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 67.2%

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

3. Applied rewrites 67.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 59.6%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot x\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot x\right)\right)} \cdot x\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(\left(c \cdot \color{blue}{\left(c \cdot x\right)}\right) \cdot x\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot \left(\left(c \cdot x\right) \cdot x\right)\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(c \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot x\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \color{blue}{\left(\left(\left(c \cdot x\right) \cdot x\right) \cdot c\right)}} \]

   11. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot x\right)\right) \cdot c}} \]

8. Applied rewrites 76.1%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot s\right) \cdot \left(\left(\left(s \cdot x\right) \cdot c\right) \cdot x\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(\left(s \cdot x\right) \cdot c\right)} \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot x\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot x\right)} \]

   6. lower-*.f64 76.8%

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot x\right)} \]

10. Applied rewrites 76.8%

    \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot x\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow c 2) (* (* x (pow s 2)) x))))