mixedcos

Percentage Accurate: 67.2% → 99.1%

Time: 3.3s

Alternatives: 7

Speedup: 2.4×

• 67.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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 (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \left|x\right| \cdot t\_0\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := \left(t\_2 \cdot \left|x\right|\right) \cdot t\_0\\ \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(\left(t\_2 \cdot t\_1\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_3}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* (fabs x) t_0))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* (* t_2 (fabs x)) t_0)))
  (if (<= (fabs x) 4e-53)
    (/ 1.0 (* t_2 (* (* t_2 t_1) t_1)))
    (/ (/ (cos (+ (fabs x) (fabs x))) t_3) t_3))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fabs(x) * t_0;
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = (t_2 * fabs(x)) * t_0;
	double tmp;
	if (fabs(x) <= 4e-53) {
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	} else {
		tmp = (cos((fabs(x) + fabs(x))) / t_3) / t_3;
	}
	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 = abs(x) * t_0
    t_2 = fmin(abs(c), abs(s))
    t_3 = (t_2 * abs(x)) * t_0
    if (abs(x) <= 4d-53) then
        tmp = 1.0d0 / (t_2 * ((t_2 * t_1) * t_1))
    else
        tmp = (cos((abs(x) + abs(x))) / t_3) / t_3
    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.abs(x) * t_0;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = (t_2 * Math.abs(x)) * t_0;
	double tmp;
	if (Math.abs(x) <= 4e-53) {
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	} else {
		tmp = (Math.cos((Math.abs(x) + Math.abs(x))) / t_3) / t_3;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = math.fabs(x) * t_0
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = (t_2 * math.fabs(x)) * t_0
	tmp = 0
	if math.fabs(x) <= 4e-53:
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1))
	else:
		tmp = (math.cos((math.fabs(x) + math.fabs(x))) / t_3) / t_3
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(abs(x) * t_0)
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(Float64(t_2 * abs(x)) * t_0)
	tmp = 0.0
	if (abs(x) <= 4e-53)
		tmp = Float64(1.0 / Float64(t_2 * Float64(Float64(t_2 * t_1) * t_1)));
	else
		tmp = Float64(Float64(cos(Float64(abs(x) + abs(x))) / t_3) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = abs(x) * t_0;
	t_2 = min(abs(c), abs(s));
	t_3 = (t_2 * abs(x)) * t_0;
	tmp = 0.0;
	if (abs(x) <= 4e-53)
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	else
		tmp = (cos((abs(x) + abs(x))) / t_3) / t_3;
	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[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 4e-53], N[(1.0 / N[(t$95$2 * N[(N[(t$95$2 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000001e-53

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.7%

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

   6. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 77.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.4%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.4%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 77.4%

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

   8. Applied rewrites 77.4%

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

   if 4.0000000000000001e-53 < 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

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

      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 78.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot x}}{s}}{s \cdot x}} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 93.7%

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

   7. Applied rewrites 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 97.2%

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

   9. Applied rewrites 97.2%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \left|x\right| \cdot t\_0\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(\left(t\_2 \cdot t\_1\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_3 \cdot \left(t\_0 \cdot \left(t\_3 \cdot t\_0\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* (fabs x) t_0))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* t_2 (fabs x))))
  (if (<= (fabs x) 2e-13)
    (/ 1.0 (* t_2 (* (* t_2 t_1) t_1)))
    (/ (cos (+ (fabs x) (fabs x))) (* t_3 (* t_0 (* t_3 t_0)))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fabs(x) * t_0;
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * fabs(x);
	double tmp;
	if (fabs(x) <= 2e-13) {
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	} else {
		tmp = cos((fabs(x) + fabs(x))) / (t_3 * (t_0 * (t_3 * 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 = abs(x) * t_0
    t_2 = fmin(abs(c), abs(s))
    t_3 = t_2 * abs(x)
    if (abs(x) <= 2d-13) then
        tmp = 1.0d0 / (t_2 * ((t_2 * t_1) * t_1))
    else
        tmp = cos((abs(x) + abs(x))) / (t_3 * (t_0 * (t_3 * 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.abs(x) * t_0;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = t_2 * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 2e-13) {
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / (t_3 * (t_0 * (t_3 * t_0)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = math.fabs(x) * t_0
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = t_2 * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 2e-13:
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1))
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / (t_3 * (t_0 * (t_3 * t_0)))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(abs(x) * t_0)
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * abs(x))
	tmp = 0.0
	if (abs(x) <= 2e-13)
		tmp = Float64(1.0 / Float64(t_2 * Float64(Float64(t_2 * t_1) * t_1)));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(t_3 * Float64(t_0 * Float64(t_3 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = abs(x) * t_0;
	t_2 = min(abs(c), abs(s));
	t_3 = t_2 * abs(x);
	tmp = 0.0;
	if (abs(x) <= 2e-13)
		tmp = 1.0 / (t_2 * ((t_2 * t_1) * t_1));
	else
		tmp = cos((abs(x) + abs(x))) / (t_3 * (t_0 * (t_3 * 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[(N[Abs[x], $MachinePrecision] * t$95$0), $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]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-13], N[(1.0 / N[(t$95$2 * N[(N[(t$95$2 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 * N[(t$95$0 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-13

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.7%

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

   6. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 77.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.4%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.4%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 77.4%

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

   8. Applied rewrites 77.4%

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

   if 2.0000000000000001e-13 < 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

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

      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 78.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot x}}{s}}{s \cdot x}} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 93.7%

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

   7. Applied rewrites 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.2%

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

   9. Applied rewrites 93.2%

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

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \left|x\right| \cdot t\_0\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot t\_1\\ \mathbf{if}\;\left|x\right| \leq 2.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(t\_3 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_0 \cdot \left(\left(t\_3 \cdot \left|x\right|\right) \cdot t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* (fabs x) t_0))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* t_2 t_1)))
  (if (<= (fabs x) 2.2e-55)
    (/ 1.0 (* t_2 (* t_3 t_1)))
    (/ (cos (+ (fabs x) (fabs x))) (* t_0 (* (* t_3 (fabs x)) t_2))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fabs(x) * t_0;
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if (fabs(x) <= 2.2e-55) {
		tmp = 1.0 / (t_2 * (t_3 * t_1));
	} else {
		tmp = cos((fabs(x) + fabs(x))) / (t_0 * ((t_3 * fabs(x)) * t_2));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = abs(x) * t_0
    t_2 = fmin(abs(c), abs(s))
    t_3 = t_2 * t_1
    if (abs(x) <= 2.2d-55) then
        tmp = 1.0d0 / (t_2 * (t_3 * t_1))
    else
        tmp = cos((abs(x) + abs(x))) / (t_0 * ((t_3 * abs(x)) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = Math.abs(x) * t_0;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if (Math.abs(x) <= 2.2e-55) {
		tmp = 1.0 / (t_2 * (t_3 * t_1));
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / (t_0 * ((t_3 * Math.abs(x)) * t_2));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = math.fabs(x) * t_0
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = t_2 * t_1
	tmp = 0
	if math.fabs(x) <= 2.2e-55:
		tmp = 1.0 / (t_2 * (t_3 * t_1))
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / (t_0 * ((t_3 * math.fabs(x)) * t_2))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(abs(x) * t_0)
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * t_1)
	tmp = 0.0
	if (abs(x) <= 2.2e-55)
		tmp = Float64(1.0 / Float64(t_2 * Float64(t_3 * t_1)));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(t_0 * Float64(Float64(t_3 * abs(x)) * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = abs(x) * t_0;
	t_2 = min(abs(c), abs(s));
	t_3 = t_2 * t_1;
	tmp = 0.0;
	if (abs(x) <= 2.2e-55)
		tmp = 1.0 / (t_2 * (t_3 * t_1));
	else
		tmp = cos((abs(x) + abs(x))) / (t_0 * ((t_3 * abs(x)) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * t$95$0), $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], 2.2e-55], N[(1.0 / N[(t$95$2 * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(N[(t$95$3 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e-55

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.7%

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

   6. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 77.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.4%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.4%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 77.4%

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

   8. Applied rewrites 77.4%

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

   if 2.2e-55 < 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

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

      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 78.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot x}}{s}}{s \cdot x}} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.9%

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 90.6%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.6%

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

   7. Applied rewrites 90.6%

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

Alternative 4: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := x \cdot t\_0\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot t\_1\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{t\_2}^{2} \cdot \left(\left(x \cdot {t\_0}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{1 + -2 \cdot {x}^{2}}{t\_0 \cdot \left(\left(t\_3 \cdot x\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(t\_3 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* x t_0))
       (t_2 (fmin (fabs c) (fabs s)))
       (t_3 (* t_2 t_1)))
  (if (<=
       (/ (cos (* 2.0 x)) (* (pow t_2 2.0) (* (* x (pow t_0 2.0)) x)))
       -1e-171)
    (/ (+ 1.0 (* -2.0 (pow x 2.0))) (* t_0 (* (* t_3 x) t_2)))
    (/ 1.0 (* t_2 (* t_3 t_1))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = x * t_0;
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if ((cos((2.0 * x)) / (pow(t_2, 2.0) * ((x * pow(t_0, 2.0)) * x))) <= -1e-171) {
		tmp = (1.0 + (-2.0 * pow(x, 2.0))) / (t_0 * ((t_3 * x) * t_2));
	} else {
		tmp = 1.0 / (t_2 * (t_3 * 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) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = x * t_0
    t_2 = fmin(abs(c), abs(s))
    t_3 = t_2 * t_1
    if ((cos((2.0d0 * x)) / ((t_2 ** 2.0d0) * ((x * (t_0 ** 2.0d0)) * x))) <= (-1d-171)) then
        tmp = (1.0d0 + ((-2.0d0) * (x ** 2.0d0))) / (t_0 * ((t_3 * x) * t_2))
    else
        tmp = 1.0d0 / (t_2 * (t_3 * 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 = x * t_0;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(t_2, 2.0) * ((x * Math.pow(t_0, 2.0)) * x))) <= -1e-171) {
		tmp = (1.0 + (-2.0 * Math.pow(x, 2.0))) / (t_0 * ((t_3 * x) * t_2));
	} else {
		tmp = 1.0 / (t_2 * (t_3 * t_1));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = x * t_0
	t_2 = fmin(math.fabs(c), math.fabs(s))
	t_3 = t_2 * t_1
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(t_2, 2.0) * ((x * math.pow(t_0, 2.0)) * x))) <= -1e-171:
		tmp = (1.0 + (-2.0 * math.pow(x, 2.0))) / (t_0 * ((t_3 * x) * t_2))
	else:
		tmp = 1.0 / (t_2 * (t_3 * t_1))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(x * t_0)
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * t_1)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((t_2 ^ 2.0) * Float64(Float64(x * (t_0 ^ 2.0)) * x))) <= -1e-171)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * (x ^ 2.0))) / Float64(t_0 * Float64(Float64(t_3 * x) * t_2)));
	else
		tmp = Float64(1.0 / Float64(t_2 * Float64(t_3 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = x * t_0;
	t_2 = min(abs(c), abs(s));
	t_3 = t_2 * t_1;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((t_2 ^ 2.0) * ((x * (t_0 ^ 2.0)) * x))) <= -1e-171)
		tmp = (1.0 + (-2.0 * (x ^ 2.0))) / (t_0 * ((t_3 * x) * t_2));
	else
		tmp = 1.0 / (t_2 * (t_3 * 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[(x * t$95$0), $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[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(x * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-171], N[(N[(1.0 + N[(-2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(t$95$3 * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999998e-172

   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 78.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot x}}{s}}{s \cdot x}} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.9%

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 90.6%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.6%

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 59.2%

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

   10. Applied rewrites 59.2%

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

   if -9.9999999999999998e-172 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.7%

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

   6. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 77.4%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.4%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.4%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 77.4%

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

   8. Applied rewrites 77.4%

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

Alternative 5: 79.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (* x (fmax (fabs c) (fabs s)))))
  (/ 1.0 (* t_0 (* (* t_0 t_1) t_1)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmin(abs(c), abs(s))
    t_1 = x * fmax(abs(c), abs(s))
    code = 1.0d0 / (t_0 * ((t_0 * t_1) * t_1))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = x * max(abs(c), abs(s));
	tmp = 1.0 / (t_0 * ((t_0 * t_1) * t_1));
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $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. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 66.7%

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

6. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 77.4%

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 77.4%

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 77.4%

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 77.4%

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

8. Applied rewrites 77.4%

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

Alternative 6: 75.9% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s))) (t_1 (fmax (fabs c) (fabs s))))
  (/ 1.0 (* t_0 (* t_1 (* (* (* x 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_0 * (t_1 * (((x * 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_0 * (t_1 * (((x * 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_0 * (t_1 * (((x * 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_0 * (t_1 * (((x * 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(t_0 * Float64(t_1 * Float64(Float64(Float64(x * 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_0 * (t_1 * (((x * 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[(1.0 / N[(t$95$0 * N[(t$95$1 * N[(N[(N[(x * t$95$1), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $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. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 66.7%

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

6. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 75.6%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 75.6%

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

8. Applied rewrites 75.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 73.1%

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

10. Applied rewrites 73.1%

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

Alternative 7: 75.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin c (fabs s))) (t_1 (fmax c (fabs s))))
  (/ 1.0 (* t_0 (* t_1 (* (* x t_1) (* t_0 x)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = min(c, abs(s));
	t_1 = max(c, abs(s));
	tmp = 1.0 / (t_0 * (t_1 * ((x * t_1) * (t_0 * x))));
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[c, N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[c, N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(t$95$0 * N[(t$95$1 * N[(N[(x * t$95$1), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 66.7%

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

6. Applied rewrites 66.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 75.6%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 75.6%

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

8. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2025212 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))