mixedcos

Percentage Accurate: 65.9% → 99.4%

Time: 2.9s

Alternatives: 10

Speedup: 2.1×

• 68.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (/ (cos (* 2 x)) (* (pow c 2) (* (* x (pow s 2)) x))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2], $MachinePrecision] * N[(N[(x * N[Power[s, 2], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x c s)
  :precision binary64
  (/ (cos (* 2 x)) (* (pow c 2) (* (* x (pow s 2)) x))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2], $MachinePrecision] * N[(N[(x * N[Power[s, 2], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot t\_1\\ t_3 := \left(t\_1 \cdot \left|x\right|\right) \cdot \left(-t\_0\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{2993155353253689}{2993155353253689176481146537402947624255349848014848}:\\ \;\;\;\;\frac{1}{t\_3 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_2}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* (* t_0 (fabs x)) t_1))
       (t_3 (* (* t_1 (fabs x)) (- t_0))))
  (if (<=
       (fabs x)
       2993155353253689/2993155353253689176481146537402947624255349848014848)
    (/ 1 (* t_3 t_3))
    (/ (/ (cos (+ (fabs x) (fabs x))) t_2) t_2))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = (t_0 * fabs(x)) * t_1;
	double t_3 = (t_1 * fabs(x)) * -t_0;
	double tmp;
	if (fabs(x) <= 1e-36) {
		tmp = 1.0 / (t_3 * t_3);
	} else {
		tmp = (cos((fabs(x) + fabs(x))) / t_2) / t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = (t_0 * abs(x)) * t_1
    t_3 = (t_1 * abs(x)) * -t_0
    if (abs(x) <= 1d-36) then
        tmp = 1.0d0 / (t_3 * t_3)
    else
        tmp = (cos((abs(x) + abs(x))) / t_2) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = (t_0 * Math.abs(x)) * t_1;
	double t_3 = (t_1 * Math.abs(x)) * -t_0;
	double tmp;
	if (Math.abs(x) <= 1e-36) {
		tmp = 1.0 / (t_3 * t_3);
	} else {
		tmp = (Math.cos((Math.abs(x) + Math.abs(x))) / t_2) / t_2;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = (t_0 * math.fabs(x)) * t_1
	t_3 = (t_1 * math.fabs(x)) * -t_0
	tmp = 0
	if math.fabs(x) <= 1e-36:
		tmp = 1.0 / (t_3 * t_3)
	else:
		tmp = (math.cos((math.fabs(x) + math.fabs(x))) / t_2) / t_2
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(Float64(t_0 * abs(x)) * t_1)
	t_3 = Float64(Float64(t_1 * abs(x)) * Float64(-t_0))
	tmp = 0.0
	if (abs(x) <= 1e-36)
		tmp = Float64(1.0 / Float64(t_3 * t_3));
	else
		tmp = Float64(Float64(cos(Float64(abs(x) + abs(x))) / t_2) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = (t_0 * abs(x)) * t_1;
	t_3 = (t_1 * abs(x)) * -t_0;
	tmp = 0.0;
	if (abs(x) <= 1e-36)
		tmp = 1.0 / (t_3 * t_3);
	else
		tmp = (cos((abs(x) + abs(x))) / t_2) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2993155353253689/2993155353253689176481146537402947624255349848014848], N[(1 / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e-37

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 77.9%

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

   8. Applied rewrites 77.9%

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

   if 9.9999999999999994e-37 < x

   1. Initial program 65.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.9%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.0%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.0%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. cos-sum N/A

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

      5. cos-2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. unswap-sqr N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.8%

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

   9. Applied rewrites 94.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 97.3%

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

   11. Applied rewrites 97.3%

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

Alternative 2: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_1 \cdot \left|x\right|\right) \cdot \left(-t\_0\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{2993155353253689}{5986310706507378352962293074805895248510699696029696}:\\ \;\;\;\;\frac{1}{t\_2 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(\left|x\right| \cdot t\_0\right) \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(t\_0 \cdot \left|x\right|\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* (* t_1 (fabs x)) (- t_0))))
  (if (<=
       (fabs x)
       2993155353253689/5986310706507378352962293074805895248510699696029696)
    (/ 1 (* t_2 t_2))
    (/
     (cos (+ (fabs x) (fabs x)))
     (* (* (fabs x) t_0) (* t_1 (* t_1 (* t_0 (fabs x)))))))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = (t_1 * fabs(x)) * -t_0;
	double tmp;
	if (fabs(x) <= 5e-37) {
		tmp = 1.0 / (t_2 * t_2);
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((fabs(x) * t_0) * (t_1 * (t_1 * (t_0 * fabs(x)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = (t_1 * abs(x)) * -t_0
    if (abs(x) <= 5d-37) then
        tmp = 1.0d0 / (t_2 * t_2)
    else
        tmp = cos((abs(x) + abs(x))) / ((abs(x) * t_0) * (t_1 * (t_1 * (t_0 * abs(x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = (t_1 * Math.abs(x)) * -t_0;
	double tmp;
	if (Math.abs(x) <= 5e-37) {
		tmp = 1.0 / (t_2 * t_2);
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((Math.abs(x) * t_0) * (t_1 * (t_1 * (t_0 * Math.abs(x)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = (t_1 * math.fabs(x)) * -t_0
	tmp = 0
	if math.fabs(x) <= 5e-37:
		tmp = 1.0 / (t_2 * t_2)
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((math.fabs(x) * t_0) * (t_1 * (t_1 * (t_0 * math.fabs(x)))))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(Float64(t_1 * abs(x)) * Float64(-t_0))
	tmp = 0.0
	if (abs(x) <= 5e-37)
		tmp = Float64(1.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(abs(x) * t_0) * Float64(t_1 * Float64(t_1 * Float64(t_0 * abs(x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = (t_1 * abs(x)) * -t_0;
	tmp = 0.0;
	if (abs(x) <= 5e-37)
		tmp = 1.0 / (t_2 * t_2);
	else
		tmp = cos((abs(x) + abs(x))) / ((abs(x) * t_0) * (t_1 * (t_1 * (t_0 * abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2993155353253689/5986310706507378352962293074805895248510699696029696], N[(1 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e-37

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 77.9%

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

   8. Applied rewrites 77.9%

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

   if 4.9999999999999997e-37 < x

   1. Initial program 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.0%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.0%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.0%

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

   3. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.7%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.7%

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

   5. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 92.7%

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

   7. Applied rewrites 92.7%

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

Alternative 3: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_1 \cdot \left|x\right|\\ t_3 := t\_2 \cdot \left(-t\_0\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{7482888383134223}{748288838313422294120286634350736906063837462003712}:\\ \;\;\;\;\frac{1}{t\_3 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_1 \cdot \left(t\_2 \cdot \left(t\_0 \cdot \left|x\right|\right)\right)\right) \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* t_1 (fabs x)))
       (t_3 (* t_2 (- t_0))))
  (if (<=
       (fabs x)
       7482888383134223/748288838313422294120286634350736906063837462003712)
    (/ 1 (* t_3 t_3))
    (/
     (cos (+ (fabs x) (fabs x)))
     (* (* t_1 (* t_2 (* t_0 (fabs x)))) t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = t_1 * fabs(x);
	double t_3 = t_2 * -t_0;
	double tmp;
	if (fabs(x) <= 1e-35) {
		tmp = 1.0 / (t_3 * t_3);
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((t_1 * (t_2 * (t_0 * fabs(x)))) * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = t_1 * abs(x)
    t_3 = t_2 * -t_0
    if (abs(x) <= 1d-35) then
        tmp = 1.0d0 / (t_3 * t_3)
    else
        tmp = cos((abs(x) + abs(x))) / ((t_1 * (t_2 * (t_0 * abs(x)))) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = t_1 * Math.abs(x);
	double t_3 = t_2 * -t_0;
	double tmp;
	if (Math.abs(x) <= 1e-35) {
		tmp = 1.0 / (t_3 * t_3);
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_1 * (t_2 * (t_0 * Math.abs(x)))) * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = t_1 * math.fabs(x)
	t_3 = t_2 * -t_0
	tmp = 0
	if math.fabs(x) <= 1e-35:
		tmp = 1.0 / (t_3 * t_3)
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_1 * (t_2 * (t_0 * math.fabs(x)))) * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(t_1 * abs(x))
	t_3 = Float64(t_2 * Float64(-t_0))
	tmp = 0.0
	if (abs(x) <= 1e-35)
		tmp = Float64(1.0 / Float64(t_3 * t_3));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_1 * Float64(t_2 * Float64(t_0 * abs(x)))) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = t_1 * abs(x);
	t_3 = t_2 * -t_0;
	tmp = 0.0;
	if (abs(x) <= 1e-35)
		tmp = 1.0 / (t_3 * t_3);
	else
		tmp = cos((abs(x) + abs(x))) / ((t_1 * (t_2 * (t_0 * abs(x)))) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * (-t$95$0)), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 7482888383134223/748288838313422294120286634350736906063837462003712], N[(1 / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$1 * N[(t$95$2 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-35

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 77.9%

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

   8. Applied rewrites 77.9%

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

   if 1e-35 < x

   1. Initial program 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.1%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 77.1%

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 77.1%

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

   3. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.6%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.6%

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

   5. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 89.6%

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

   7. Applied rewrites 89.6%

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

Alternative 4: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_1 \cdot \left|x\right|\right) \cdot \left(-t\_0\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{940834429856889}{618970019642690137449562112}:\\ \;\;\;\;\frac{1}{t\_2 \cdot t\_2}\\ \mathbf{elif}\;\left|x\right| \leq 274999999999999995274917990641983180319600859774341068865301343307866076067129259703961489954509259284386354608521075666727825095781944535915128938299392:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_0 \cdot \left(t\_0 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left(-t\_1\right)\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* (* t_1 (fabs x)) (- t_0))))
  (if (<= (fabs x) 940834429856889/618970019642690137449562112)
    (/ 1 (* t_2 t_2))
    (if (<=
         (fabs x)
         274999999999999995274917990641983180319600859774341068865301343307866076067129259703961489954509259284386354608521075666727825095781944535915128938299392)
      (/
       (cos (+ (fabs x) (fabs x)))
       (* (* t_0 (* t_0 (* (fabs x) (fabs x)))) (* t_1 t_1)))
      (/ 1 (pow (* (* t_0 (fabs x)) (- t_1)) 2))))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = (t_1 * fabs(x)) * -t_0;
	double tmp;
	if (fabs(x) <= 1.52e-12) {
		tmp = 1.0 / (t_2 * t_2);
	} else if (fabs(x) <= 2.75e+152) {
		tmp = cos((fabs(x) + fabs(x))) / ((t_0 * (t_0 * (fabs(x) * fabs(x)))) * (t_1 * t_1));
	} else {
		tmp = 1.0 / pow(((t_0 * fabs(x)) * -t_1), 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = (t_1 * abs(x)) * -t_0
    if (abs(x) <= 1.52d-12) then
        tmp = 1.0d0 / (t_2 * t_2)
    else if (abs(x) <= 2.75d+152) then
        tmp = cos((abs(x) + abs(x))) / ((t_0 * (t_0 * (abs(x) * abs(x)))) * (t_1 * t_1))
    else
        tmp = 1.0d0 / (((t_0 * abs(x)) * -t_1) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = (t_1 * Math.abs(x)) * -t_0;
	double tmp;
	if (Math.abs(x) <= 1.52e-12) {
		tmp = 1.0 / (t_2 * t_2);
	} else if (Math.abs(x) <= 2.75e+152) {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_0 * (t_0 * (Math.abs(x) * Math.abs(x)))) * (t_1 * t_1));
	} else {
		tmp = 1.0 / Math.pow(((t_0 * Math.abs(x)) * -t_1), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = (t_1 * math.fabs(x)) * -t_0
	tmp = 0
	if math.fabs(x) <= 1.52e-12:
		tmp = 1.0 / (t_2 * t_2)
	elif math.fabs(x) <= 2.75e+152:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_0 * (t_0 * (math.fabs(x) * math.fabs(x)))) * (t_1 * t_1))
	else:
		tmp = 1.0 / math.pow(((t_0 * math.fabs(x)) * -t_1), 2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(Float64(t_1 * abs(x)) * Float64(-t_0))
	tmp = 0.0
	if (abs(x) <= 1.52e-12)
		tmp = Float64(1.0 / Float64(t_2 * t_2));
	elseif (abs(x) <= 2.75e+152)
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_0 * Float64(t_0 * Float64(abs(x) * abs(x)))) * Float64(t_1 * t_1)));
	else
		tmp = Float64(1.0 / (Float64(Float64(t_0 * abs(x)) * Float64(-t_1)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = (t_1 * abs(x)) * -t_0;
	tmp = 0.0;
	if (abs(x) <= 1.52e-12)
		tmp = 1.0 / (t_2 * t_2);
	elseif (abs(x) <= 2.75e+152)
		tmp = cos((abs(x) + abs(x))) / ((t_0 * (t_0 * (abs(x) * abs(x)))) * (t_1 * t_1));
	else
		tmp = 1.0 / (((t_0 * abs(x)) * -t_1) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 940834429856889/618970019642690137449562112], N[(1 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 274999999999999995274917990641983180319600859774341068865301343307866076067129259703961489954509259284386354608521075666727825095781944535915128938299392], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$0 * N[(t$95$0 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[Power[N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.52e-12

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 77.9%

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

   8. Applied rewrites 77.9%

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

   if 1.52e-12 < x < 2.75e152

   1. Initial program 65.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.9%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.0%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.0%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

   if 2.75e152 < x

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. pow-prod-down N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 78.1%

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

   8. Applied rewrites 78.1%

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

Alternative 5: 86.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := s \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left(-c\right)\\ \mathbf{if}\;\left|x\right| \leq \frac{822752278660603}{822752278660603021077484591278675252491367932816789931674304512}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(c \cdot \left(s \cdot c\right)\right) \cdot \left(t\_0 \cdot \left|x\right|\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (* s (fabs x))) (t_1 (* t_0 (- c))))
  (if (<=
       (fabs x)
       822752278660603/822752278660603021077484591278675252491367932816789931674304512)
    (/ 1 (* t_1 t_1))
    (/
     (cos (+ (fabs x) (fabs x)))
     (* (* c (* s c)) (* t_0 (fabs x)))))))

C

double code(double x, double c, double s) {
	double t_0 = s * fabs(x);
	double t_1 = t_0 * -c;
	double tmp;
	if (fabs(x) <= 1e-48) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = cos((fabs(x) + fabs(x))) / ((c * (s * c)) * (t_0 * fabs(x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * abs(x)
    t_1 = t_0 * -c
    if (abs(x) <= 1d-48) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = cos((abs(x) + abs(x))) / ((c * (s * c)) * (t_0 * abs(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = s * Math.abs(x);
	double t_1 = t_0 * -c;
	double tmp;
	if (Math.abs(x) <= 1e-48) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((c * (s * c)) * (t_0 * Math.abs(x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = s * math.fabs(x)
	t_1 = t_0 * -c
	tmp = 0
	if math.fabs(x) <= 1e-48:
		tmp = 1.0 / (t_1 * t_1)
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((c * (s * c)) * (t_0 * math.fabs(x)))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(s * abs(x))
	t_1 = Float64(t_0 * Float64(-c))
	tmp = 0.0
	if (abs(x) <= 1e-48)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(c * Float64(s * c)) * Float64(t_0 * abs(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = s * abs(x);
	t_1 = t_0 * -c;
	tmp = 0.0;
	if (abs(x) <= 1e-48)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = cos((abs(x) + abs(x))) / ((c * (s * c)) * (t_0 * abs(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * (-c)), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 822752278660603/822752278660603021077484591278675252491367932816789931674304512], N[(1 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999997e-49

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 77.9%

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

   8. Applied rewrites 77.9%

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

   if 9.9999999999999997e-49 < x

   1. Initial program 65.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.9%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.0%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.0%

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 66.0%

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

   3. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 66.6%

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 80.9%

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

Alternative 6: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.1%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. *-commutative N/A

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

   6. lower-*.f64 58.1%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 54.7%

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

6. Applied rewrites 54.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. unswap-sqr N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 77.9%

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

8. Applied rewrites 77.9%

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

Alternative 7: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;{t\_1}^{2} \leq \frac{4586997231980143}{4586997231980143023221641790604173881593129978336562247475177678773845752176969616140037106220251373109248}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(x \cdot \left(t\_2 \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(t\_1 \cdot t\_1\right) \cdot x\right) \cdot t\_0\right) \cdot \left(t\_0 \cdot x\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (fmin (fabs c) (fabs s)))
       (t_2 (* t_0 t_1)))
  (if (<=
       (pow t_1 2)
       4586997231980143/4586997231980143023221641790604173881593129978336562247475177678773845752176969616140037106220251373109248)
    (/ 1 (* t_2 (* x (* t_2 x))))
    (/ 1 (* (* (* (* t_1 t_1) x) t_0) (* t_0 x))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double t_2 = t_0 * t_1;
	double tmp;
	if (pow(t_1, 2.0) <= 1e-90) {
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	} else {
		tmp = 1.0 / ((((t_1 * t_1) * x) * t_0) * (t_0 * x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = t_0 * t_1
    if ((t_1 ** 2.0d0) <= 1d-90) then
        tmp = 1.0d0 / (t_2 * (x * (t_2 * x)))
    else
        tmp = 1.0d0 / ((((t_1 * t_1) * x) * t_0) * (t_0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	double t_2 = t_0 * t_1;
	double tmp;
	if (Math.pow(t_1, 2.0) <= 1e-90) {
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	} else {
		tmp = 1.0 / ((((t_1 * t_1) * x) * t_0) * (t_0 * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	t_2 = t_0 * t_1
	tmp = 0
	if math.pow(t_1, 2.0) <= 1e-90:
		tmp = 1.0 / (t_2 * (x * (t_2 * x)))
	else:
		tmp = 1.0 / ((((t_1 * t_1) * x) * t_0) * (t_0 * x))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if ((t_1 ^ 2.0) <= 1e-90)
		tmp = Float64(1.0 / Float64(t_2 * Float64(x * Float64(t_2 * x))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(t_1 * t_1) * x) * t_0) * Float64(t_0 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if ((t_1 ^ 2.0) <= 1e-90)
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	else
		tmp = 1.0 / ((((t_1 * t_1) * x) * t_0) * (t_0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[t$95$1, 2], $MachinePrecision], 4586997231980143/4586997231980143023221641790604173881593129978336562247475177678773845752176969616140037106220251373109248], N[(1 / N[(t$95$2 * N[(x * N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 9.9999999999999999e-91

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unswap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 75.0%

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

      26. associate-*r* N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

   8. Applied rewrites 75.7%

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

   if 9.9999999999999999e-91 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative 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. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 66.4%

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

   6. Applied rewrites 66.4%

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

Alternative 8: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_1 \cdot t\_0\\ \mathbf{if}\;t\_0 \leq \frac{2758407706096627}{81129638414606681695789005144064}:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(x \cdot \left(t\_2 \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(\left(\left(t\_1 \cdot x\right) \cdot t\_1\right) \cdot \left(t\_0 \cdot x\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmin (fabs c) (fabs s)))
       (t_1 (fmax (fabs c) (fabs s)))
       (t_2 (* t_1 t_0)))
  (if (<= t_0 2758407706096627/81129638414606681695789005144064)
    (/ 1 (* t_2 (* x (* t_2 x))))
    (/ 1 (* t_0 (* (* (* t_1 x) t_1) (* t_0 x)))))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = t_1 * t_0;
	double tmp;
	if (t_0 <= 3.4e-17) {
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	} else {
		tmp = 1.0 / (t_0 * (((t_1 * x) * t_1) * (t_0 * x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = t_1 * t_0
    if (t_0 <= 3.4d-17) then
        tmp = 1.0d0 / (t_2 * (x * (t_2 * x)))
    else
        tmp = 1.0d0 / (t_0 * (((t_1 * x) * t_1) * (t_0 * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = t_1 * t_0;
	double tmp;
	if (t_0 <= 3.4e-17) {
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	} else {
		tmp = 1.0 / (t_0 * (((t_1 * x) * t_1) * (t_0 * x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = t_1 * t_0
	tmp = 0
	if t_0 <= 3.4e-17:
		tmp = 1.0 / (t_2 * (x * (t_2 * x)))
	else:
		tmp = 1.0 / (t_0 * (((t_1 * x) * t_1) * (t_0 * x)))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(t_1 * t_0)
	tmp = 0.0
	if (t_0 <= 3.4e-17)
		tmp = Float64(1.0 / Float64(t_2 * Float64(x * Float64(t_2 * x))));
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(Float64(t_1 * x) * t_1) * Float64(t_0 * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = t_1 * t_0;
	tmp = 0.0;
	if (t_0 <= 3.4e-17)
		tmp = 1.0 / (t_2 * (x * (t_2 * x)));
	else
		tmp = 1.0 / (t_0 * (((t_1 * x) * t_1) * (t_0 * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, 2758407706096627/81129638414606681695789005144064], N[(1 / N[(t$95$2 * N[(x * N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(t$95$0 * N[(N[(N[(t$95$1 * x), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 3.3999999999999998e-17

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unswap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 75.0%

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

      26. associate-*r* N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

   8. Applied rewrites 75.7%

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

   if 3.3999999999999998e-17 < c

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   8. Applied rewrites 71.7%

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

Alternative 9: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot x\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \mathbf{if}\;t\_0 \leq 460000000000000022101119877342656515916467987006775686626865640412119103751870846467046189291054240972989870508637752971162791533089976828878079167552007296402010877485466082369922318385290320066955959683353273171968:\\ \;\;\;\;\frac{1}{t\_2 \cdot \left(\left(t\_1 \cdot t\_0\right) \cdot \left(t\_2 \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t\_2 \cdot \left(t\_0 \cdot t\_2\right)\right) \cdot \left(t\_1 \cdot x\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
  :precision binary64
  (let* ((t_0 (fmax (fabs c) (fabs s)))
       (t_1 (* t_0 x))
       (t_2 (fmin (fabs c) (fabs s))))
  (if (<=
       t_0
       460000000000000022101119877342656515916467987006775686626865640412119103751870846467046189291054240972989870508637752971162791533089976828878079167552007296402010877485466082369922318385290320066955959683353273171968)
    (/ 1 (* t_2 (* (* t_1 t_0) (* t_2 x))))
    (/ 1 (* (* t_2 (* t_0 t_2)) (* t_1 x))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(fabs(c), fabs(s));
	double tmp;
	if (t_0 <= 4.6e+215) {
		tmp = 1.0 / (t_2 * ((t_1 * t_0) * (t_2 * x)));
	} else {
		tmp = 1.0 / ((t_2 * (t_0 * t_2)) * (t_1 * x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = t_0 * x
    t_2 = fmin(abs(c), abs(s))
    if (t_0 <= 4.6d+215) then
        tmp = 1.0d0 / (t_2 * ((t_1 * t_0) * (t_2 * x)))
    else
        tmp = 1.0d0 / ((t_2 * (t_0 * t_2)) * (t_1 * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = t_0 * x;
	double t_2 = fmin(Math.abs(c), Math.abs(s));
	double tmp;
	if (t_0 <= 4.6e+215) {
		tmp = 1.0 / (t_2 * ((t_1 * t_0) * (t_2 * x)));
	} else {
		tmp = 1.0 / ((t_2 * (t_0 * t_2)) * (t_1 * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = t_0 * x
	t_2 = fmin(math.fabs(c), math.fabs(s))
	tmp = 0
	if t_0 <= 4.6e+215:
		tmp = 1.0 / (t_2 * ((t_1 * t_0) * (t_2 * x)))
	else:
		tmp = 1.0 / ((t_2 * (t_0 * t_2)) * (t_1 * x))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * x)
	t_2 = fmin(abs(c), abs(s))
	tmp = 0.0
	if (t_0 <= 4.6e+215)
		tmp = Float64(1.0 / Float64(t_2 * Float64(Float64(t_1 * t_0) * Float64(t_2 * x))));
	else
		tmp = Float64(1.0 / Float64(Float64(t_2 * Float64(t_0 * t_2)) * Float64(t_1 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = t_0 * x;
	t_2 = min(abs(c), abs(s));
	tmp = 0.0;
	if (t_0 <= 4.6e+215)
		tmp = 1.0 / (t_2 * ((t_1 * t_0) * (t_2 * x)));
	else
		tmp = 1.0 / ((t_2 * (t_0 * t_2)) * (t_1 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 460000000000000022101119877342656515916467987006775686626865640412119103751870846467046189291054240972989870508637752971162791533089976828878079167552007296402010877485466082369922318385290320066955959683353273171968], N[(1 / N[(t$95$2 * N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(N[(t$95$2 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if s < 4.6000000000000002e215

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   8. Applied rewrites 71.7%

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

   if 4.6000000000000002e215 < s

   1. Initial program 65.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      6. lower-*.f64 58.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 54.7%

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

   6. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 69.3%

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

   8. Applied rewrites 69.3%

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

Alternative 10: 71.9% accurate, 0.8× speedup

Math

\[\frac{1}{\mathsf{min}\left(c, s\right) \cdot \left(\left(\left(\mathsf{max}\left(c, s\right) \cdot x\right) \cdot \mathsf{max}\left(c, s\right)\right) \cdot \left(\mathsf{min}\left(c, s\right) \cdot x\right)\right)} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(1 / N[(N[Min[c, s], $MachinePrecision] * N[(N[(N[(N[Max[c, s], $MachinePrecision] * x), $MachinePrecision] * N[Max[c, s], $MachinePrecision]), $MachinePrecision] * N[(N[Min[c, s], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.1%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. *-commutative N/A

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

   6. lower-*.f64 58.1%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 54.7%

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

6. Applied rewrites 54.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*l* N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 N/A

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

8. Applied rewrites 71.7%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow c 2) (* (* x (pow s 2)) x))))