mixedcos

Percentage Accurate: 67.2% → 98.7%

Time: 7.8s

Alternatives: 13

Speedup: 2.4×

• 31.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\_m\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := \left(c\_m \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{t\_1}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* s x) c_m)) (t_1 (cos (* 2.0 x))) (t_2 (* (* c_m s) x)))
   (if (<= (/ t_1 (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) INFINITY)
     (/ t_1 (* t_0 t_0))
     (/ t_1 (* t_2 t_2)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (s * x) * c_m;
	double t_1 = cos((2.0 * x));
	double t_2 = (c_m * s) * x;
	double tmp;
	if ((t_1 / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = t_1 / (t_0 * t_0);
	} else {
		tmp = t_1 / (t_2 * t_2);
	}
	return tmp;
}

Java

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

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = (s * x) * c_m
	t_1 = math.cos((2.0 * x))
	t_2 = (c_m * s) * x
	tmp = 0
	if (t_1 / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= math.inf:
		tmp = t_1 / (t_0 * t_0)
	else:
		tmp = t_1 / (t_2 * t_2)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(s * x) * c_m)
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(Float64(c_m * s) * x)
	tmp = 0.0
	if (Float64(t_1 / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= Inf)
		tmp = Float64(t_1 / Float64(t_0 * t_0));
	else
		tmp = Float64(t_1 / Float64(t_2 * t_2));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	t_0 = (s * x) * c_m;
	t_1 = cos((2.0 * x));
	t_2 = (c_m * s) * x;
	tmp = 0.0;
	if ((t_1 / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= Inf)
		tmp = t_1 / (t_0 * t_0);
	else
		tmp = t_1 / (t_2 * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 78.5

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 97.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. unpow2 N/A

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

      8. sqr-neg-rev N/A

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

      9. unswap-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 99.4

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

   8. Applied rewrites 99.4%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 66.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 93.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{1}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot x\right) \cdot c\_m}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\_m\right) \cdot s\right) \cdot c\_m\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x)))))
   (if (<= t_0 -2e-292)
     (/ (/ -2.0 c_m) (* (* s s) c_m))
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 (* (* (* (* s s) x) x) c_m)) c_m)
       (/ 1.0 (* (* (* (* s c_m) s) c_m) (* x x)))))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x));
	double tmp;
	if (t_0 <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / ((((s * s) * x) * x) * c_m)) / c_m;
	} else {
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	}
	return tmp;
}

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x));
	double tmp;
	if (t_0 <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / ((((s * s) * x) * x) * c_m)) / c_m;
	} else {
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))
	tmp = 0
	if t_0 <= -2e-292:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	elif t_0 <= math.inf:
		tmp = (1.0 / ((((s * s) * x) * x) * c_m)) / c_m
	else:
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
	tmp = 0.0
	if (t_0 <= -2e-292)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(Float64(s * s) * x) * x) * c_m)) / c_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * c_m) * s) * c_m) * Float64(x * x)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	t_0 = cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x));
	tmp = 0.0;
	if (t_0 <= -2e-292)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	elseif (t_0 <= Inf)
		tmp = (1.0 / ((((s * s) * x) * x) * c_m)) / c_m;
	else
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-292], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[(N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000001e-292

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

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

   if -2.0000000000000001e-292 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 82.6%

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

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

   1. Initial program 0.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.2

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

   6. Applied rewrites 62.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 43.7%

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

Alternative 3: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot c\_m\right)}^{-1}}{x}}{\left(c\_m \cdot s\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) -2e-292)
   (/ (/ -2.0 c_m) (* (* s s) c_m))
   (/ (/ (pow (* s c_m) -1.0) x) (* (* c_m s) x))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (pow((s * c_m), -1.0) / x) / ((c_m * s) * x);
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-2d-292)) then
        tmp = ((-2.0d0) / c_m) / ((s * s) * c_m)
    else
        tmp = (((s * c_m) ** (-1.0d0)) / x) / ((c_m * s) * x)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (Math.pow((s * c_m), -1.0) / x) / ((c_m * s) * x);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -2e-292:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	else:
		tmp = (math.pow((s * c_m), -1.0) / x) / ((c_m * s) * x)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	else
		tmp = Float64(Float64((Float64(s * c_m) ^ -1.0) / x) / Float64(Float64(c_m * s) * x));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	else
		tmp = (((s * c_m) ^ -1.0) / x) / ((c_m * s) * x);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-292], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(s * c$95$m), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000001e-292

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

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

   if -2.0000000000000001e-292 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 76.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 85.2

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

   11. Applied rewrites 85.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c}}{\left(s \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot c\right)}^{-1}}{x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(x \cdot \left(s \cdot c\_m\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) -2e-292)
   (/ (/ -2.0 c_m) (* (* s s) c_m))
   (/ 1.0 (pow (* x (* s c_m)) 2.0))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = 1.0 / pow((x * (s * c_m)), 2.0);
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-2d-292)) then
        tmp = ((-2.0d0) / c_m) / ((s * s) * c_m)
    else
        tmp = 1.0d0 / ((x * (s * c_m)) ** 2.0d0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = 1.0 / Math.pow((x * (s * c_m)), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -2e-292:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	else:
		tmp = 1.0 / math.pow((x * (s * c_m)), 2.0)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	else
		tmp = Float64(1.0 / (Float64(x * Float64(s * c_m)) ^ 2.0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	else
		tmp = 1.0 / ((x * (s * c_m)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-292], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(x * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000001e-292

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

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

   if -2.0000000000000001e-292 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 76.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.3%

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

Alternative 5: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* c_m s) x)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x)))
        -2e-292)
     (/ (/ -2.0 c_m) (* (* s s) c_m))
     (/ (/ 1.0 t_0) t_0))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (c_m * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c_m * s) * x
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-2d-292)) then
        tmp = ((-2.0d0) / c_m) / ((s * s) * c_m)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = (c_m * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = (c_m * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -2e-292:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(c_m * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	t_0 = (c_m * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-292], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000001e-292

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

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

   if -2.0000000000000001e-292 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 76.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c}}{\left(s \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\_m\right) \cdot s\right) \cdot c\_m\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) -2e-292)
   (/ (/ -2.0 c_m) (* (* s s) c_m))
   (/ 1.0 (* (* (* (* s c_m) s) c_m) (* x x)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-2d-292)) then
        tmp = ((-2.0d0) / c_m) / ((s * s) * c_m)
    else
        tmp = 1.0d0 / ((((s * c_m) * s) * c_m) * (x * x))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -2e-292) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -2e-292:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	else:
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * c_m) * s) * c_m) * Float64(x * x)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -2e-292)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	else
		tmp = 1.0 / ((((s * c_m) * s) * c_m) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-292], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000001e-292

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

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

   if -2.0000000000000001e-292 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 64.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 75.5

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

   6. Applied rewrites 75.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 69.8%

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

Alternative 7: 47.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{c\_m}}{\left(s \cdot c\_m\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) 0.0)
   (/ (/ -2.0 c_m) (* (* s s) c_m))
   (/ (/ 2.0 c_m) (* (* s c_m) s))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (2.0 / c_m) / ((s * c_m) * s);
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= 0.0d0) then
        tmp = ((-2.0d0) / c_m) / ((s * s) * c_m)
    else
        tmp = (2.0d0 / c_m) / ((s * c_m) * s)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= 0.0) {
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	} else {
		tmp = (2.0 / c_m) / ((s * c_m) * s);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= 0.0:
		tmp = (-2.0 / c_m) / ((s * s) * c_m)
	else:
		tmp = (2.0 / c_m) / ((s * c_m) * s)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= 0.0)
		tmp = Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m));
	else
		tmp = Float64(Float64(2.0 / c_m) / Float64(Float64(s * c_m) * s));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= 0.0)
		tmp = (-2.0 / c_m) / ((s * s) * c_m);
	else
		tmp = (2.0 / c_m) / ((s * c_m) * s);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / c$95$m), $MachinePrecision] / N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 0.0

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.7%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.4%

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

   if 0.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.1%

      \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 3.8%

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

   12. Applied rewrites 44.0%

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

Alternative 8: 77.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \left(s \cdot c\_m\right) \cdot x\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{t\_0}{\left(\left(x \cdot x\right) \cdot c\_m\right) \cdot \left(\left(s \cdot c\_m\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(\left(c\_m \cdot x\right) \cdot c\_m\right) \cdot x\right) \cdot s\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (* (* s c_m) x)))
   (if (<= x 1.7e-54)
     (/ (fma -2.0 (* x x) 1.0) (* t_1 t_1))
     (if (<= x 4.2e+156)
       (/ t_0 (* (* (* x x) c_m) (* (* s c_m) s)))
       (/ t_0 (* (* (* (* (* c_m x) c_m) x) s) s))))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = cos((2.0 * x));
	double t_1 = (s * c_m) * x;
	double tmp;
	if (x <= 1.7e-54) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_1 * t_1);
	} else if (x <= 4.2e+156) {
		tmp = t_0 / (((x * x) * c_m) * ((s * c_m) * s));
	} else {
		tmp = t_0 / (((((c_m * x) * c_m) * x) * s) * s);
	}
	return tmp;
}

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(Float64(s * c_m) * x)
	tmp = 0.0
	if (x <= 1.7e-54)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_1 * t_1));
	elseif (x <= 4.2e+156)
		tmp = Float64(t_0 / Float64(Float64(Float64(x * x) * c_m) * Float64(Float64(s * c_m) * s)));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(Float64(c_m * x) * c_m) * x) * s) * s));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.7e-54], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+156], N[(t$95$0 / N[(N[(N[(x * x), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(N[(N[(c$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.69999999999999994e-54

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 71.2%

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

   if 1.69999999999999994e-54 < x < 4.19999999999999963e156

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 4.19999999999999963e156 < x

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 52.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 92.0%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 79.4

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

   9. Applied rewrites 79.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot c\right) \cdot x\right) \cdot s\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot c\_m\right) \cdot x\\ t_1 := \left(c\_m \cdot s\right) \cdot x\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot x\right) \cdot c\_m\right) \cdot \left(\left(s \cdot c\_m\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* s c_m) x)) (t_1 (* (* c_m s) x)))
   (if (<= x 1.7e-54)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (if (<= x 4.2e+157)
       (/ (cos (* 2.0 x)) (* (* (* x x) c_m) (* (* s c_m) s)))
       (/ (/ 1.0 t_1) t_1)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (s * c_m) * x;
	double t_1 = (c_m * s) * x;
	double tmp;
	if (x <= 1.7e-54) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else if (x <= 4.2e+157) {
		tmp = cos((2.0 * x)) / (((x * x) * c_m) * ((s * c_m) * s));
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(s * c_m) * x)
	t_1 = Float64(Float64(c_m * s) * x)
	tmp = 0.0
	if (x <= 1.7e-54)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	elseif (x <= 4.2e+157)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64(x * x) * c_m) * Float64(Float64(s * c_m) * s)));
	else
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.7e-54], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+157], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.69999999999999994e-54

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 71.2%

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

   if 1.69999999999999994e-54 < x < 4.2e157

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 4.2e157 < x

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 52.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 91.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.9

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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\right) \cdot x\\ t_1 := \left(s \cdot c\_m\right) \cdot x\\ \mathbf{if}\;x \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(s \cdot c\_m\right) \cdot s\right) \cdot c\_m\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* c_m s) x)) (t_1 (* (* s c_m) x)))
   (if (<= x 4.8e-55)
     (/ (fma -2.0 (* x x) 1.0) (* t_1 t_1))
     (if (<= x 4.2e+157)
       (/ (cos (+ x x)) (* (* (* (* s c_m) s) c_m) (* x x)))
       (/ (/ 1.0 t_0) t_0)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (c_m * s) * x;
	double t_1 = (s * c_m) * x;
	double tmp;
	if (x <= 4.8e-55) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_1 * t_1);
	} else if (x <= 4.2e+157) {
		tmp = cos((x + x)) / ((((s * c_m) * s) * c_m) * (x * x));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(c_m * s) * x)
	t_1 = Float64(Float64(s * c_m) * x)
	tmp = 0.0
	if (x <= 4.8e-55)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_1 * t_1));
	elseif (x <= 4.2e+157)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(Float64(s * c_m) * s) * c_m) * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * c$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 4.8e-55], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+157], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(s * c$95$m), $MachinePrecision] * s), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.79999999999999983e-55

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 71.2%

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

   if 4.79999999999999983e-55 < x < 4.2e157

   1. Initial program 75.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.7

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

   8. Applied rewrites 93.7%

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

   if 4.2e157 < x

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 52.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-prod-down N/A

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

      23. lower-pow.f64 N/A

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

   6. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 91.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.9

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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(s \cdot c\right) \cdot s\right) \cdot c\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot x\right) \cdot \left(\left(c\_m \cdot c\_m\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(x \cdot \left(s \cdot c\_m\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= s 1.15e+82)
   (/ (cos (+ x x)) (* (* (* s s) x) (* (* c_m c_m) x)))
   (/ 1.0 (pow (* x (* s c_m)) 2.0))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (s <= 1.15e+82) {
		tmp = cos((x + x)) / (((s * s) * x) * ((c_m * c_m) * x));
	} else {
		tmp = 1.0 / pow((x * (s * c_m)), 2.0);
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 1.15d+82) then
        tmp = cos((x + x)) / (((s * s) * x) * ((c_m * c_m) * x))
    else
        tmp = 1.0d0 / ((x * (s * c_m)) ** 2.0d0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (s <= 1.15e+82) {
		tmp = Math.cos((x + x)) / (((s * s) * x) * ((c_m * c_m) * x));
	} else {
		tmp = 1.0 / Math.pow((x * (s * c_m)), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if s <= 1.15e+82:
		tmp = math.cos((x + x)) / (((s * s) * x) * ((c_m * c_m) * x))
	else:
		tmp = 1.0 / math.pow((x * (s * c_m)), 2.0)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (s <= 1.15e+82)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(s * s) * x) * Float64(Float64(c_m * c_m) * x)));
	else
		tmp = Float64(1.0 / (Float64(x * Float64(s * c_m)) ^ 2.0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if (s <= 1.15e+82)
		tmp = cos((x + x)) / (((s * s) * x) * ((c_m * c_m) * x));
	else
		tmp = 1.0 / ((x * (s * c_m)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[s, 1.15e+82], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision] * N[(N[(c$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(x * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.14999999999999994e82

   1. Initial program 67.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 68.6

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 68.6

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

   6. Applied rewrites 68.6%

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

   if 1.14999999999999994e82 < s

   1. Initial program 56.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(-2 \cdot x\right)}{x}}}{{\left(c \cdot s\right)}^{2} \cdot x} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cos \left(-2 \cdot x\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)}} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(-2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\cos \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\mathsf{neg}\left(2 \cdot x\right)\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      9. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right)} \cdot x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{\left(x \cdot x\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

      16. lift-/.f64 82.2

          \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

      18. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot {\left(c \cdot s\right)}^{2}}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right)} \cdot {\left(c \cdot s\right)}^{2}} \]

      20. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{x}^{2}} \cdot {\left(c \cdot s\right)}^{2}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

      22. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      23. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   6. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.5%

      \[\leadsto \frac{\color{blue}{1}}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 97.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\right) \cdot x\\ \frac{\cos \left(2 \cdot x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* (* c_m s) x))) (/ (cos (* 2.0 x)) (* t_0 t_0))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double t_0 = (c_m * s) * x;
	return cos((2.0 * x)) / (t_0 * t_0);
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (c_m * s) * x
    code = cos((2.0d0 * x)) / (t_0 * t_0)
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double t_0 = (c_m * s) * x;
	return Math.cos((2.0 * x)) / (t_0 * t_0);
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	t_0 = (c_m * s) * x
	return math.cos((2.0 * x)) / (t_0 * t_0)

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	t_0 = Float64(Float64(c_m * s) * x)
	return Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0))
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp = code(x, c_m, s)
	t_0 = (c_m * s) * x;
	tmp = cos((2.0 * x)) / (t_0 * t_0);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(c$95$m * s), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}} \]

4. Applied rewrites 88.2%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(-2 \cdot x\right)}{x}}{{\left(c \cdot s\right)}^{2} \cdot x}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(-2 \cdot x\right)}{x}}{{\left(c \cdot s\right)}^{2} \cdot x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(-2 \cdot x\right)}{x}}}{{\left(c \cdot s\right)}^{2} \cdot x} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\cos \left(-2 \cdot x\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)}} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(-2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\cos \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{\cos \color{blue}{\left(\mathsf{neg}\left(2 \cdot x\right)\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)\right)}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   9. cos-neg-rev N/A

      \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{x \cdot \left({\left(c \cdot s\right)}^{2} \cdot x\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right)} \cdot x} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{\left(x \cdot x\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

   16. lift-/.f64 76.2

       \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2} \cdot \left(x \cdot x\right)}} \]

   18. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot {\left(c \cdot s\right)}^{2}}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right)} \cdot {\left(c \cdot s\right)}^{2}} \]

   20. pow2 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{x}^{2}} \cdot {\left(c \cdot s\right)}^{2}} \]

   21. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   22. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   23. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

6. Applied rewrites 96.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}} \]

   3. lower-*.f64 96.5

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(s \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   6. lower-*.f64 96.5

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(s \cdot c\right)} \cdot x\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   9. lower-*.f64 96.5

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot x\right)}} \]

   12. lower-*.f64 96.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot x\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot x\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

   15. lower-*.f64 96.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

8. Applied rewrites 96.5%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]
9. Add Preprocessing

Alternative 13: 29.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \frac{\frac{-2}{c\_m}}{\left(s \cdot s\right) \cdot c\_m} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s) :precision binary64 (/ (/ -2.0 c_m) (* (* s s) c_m)))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	return (-2.0 / c_m) / ((s * s) * c_m);
}

Fortran

c_m =     private
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = ((-2.0d0) / c_m) / ((s * s) * c_m)
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	return (-2.0 / c_m) / ((s * s) * c_m);
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	return (-2.0 / c_m) / ((s * s) * c_m)

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	return Float64(Float64(-2.0 / c_m) / Float64(Float64(s * s) * c_m))
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp = code(x, c_m, s)
	tmp = (-2.0 / c_m) / ((s * s) * c_m);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := N[(N[(-2.0 / c$95$m), $MachinePrecision] / N[(N[(s * s), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-2 \cdot {x}^{2}}{{c}^{2} \cdot {s}^{2}}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}} \]

   2. div-add-rev N/A

      \[\leadsto \frac{\color{blue}{\frac{-2 \cdot {x}^{2} + 1}{{c}^{2} \cdot {s}^{2}}}}{{x}^{2}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{1 + -2 \cdot {x}^{2}}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{1 + -2 \cdot {x}^{2}}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + -2 \cdot {x}^{2}}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, {x}^{2}, 1\right)}}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \color{blue}{x \cdot x}, 1\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, \color{blue}{x \cdot x}, 1\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\color{blue}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\color{blue}{\left({x}^{2} \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   13. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\color{blue}{{x}^{2} \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{{x}^{2} \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{{x}^{2} \cdot \color{blue}{\left(c \cdot \left(c \cdot {s}^{2}\right)\right)}} \]

   17. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\color{blue}{\left({x}^{2} \cdot c\right) \cdot \left(c \cdot {s}^{2}\right)}} \]

5. Applied rewrites 47.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot c\right) \cdot s\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 22.8%

   \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
9. Step-by-step derivation

10. Applied rewrites 25.9%

    \[\leadsto \frac{\frac{-2}{c}}{\left(s \cdot s\right) \cdot \color{blue}{c}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))