mixedcos

Percentage Accurate: 65.8% → 97.2%

Time: 6.4s

Alternatives: 9

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: 65.8% 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: 97.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c))) (/ (cos (+ x x)) (* t_0 t_0))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	return cos((x + x)) / (t_0 * t_0);
}

Fortran

NOTE: x, c, 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, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (s * x) * c
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = (s * x) * c
	return math.cos((x + x)) / (t_0 * t_0)

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	t_0 = (s * x) * c;
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.4%

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 89.0%

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

9. Applied rewrites 97.4%

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

    1. lift-*.f64 N/A

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

    2. count-2-rev N/A

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

    3. lower-+.f64 97.4

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

11. Applied rewrites 97.4%

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

Alternative 2: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -5e-109)
     (/ -2.0 (* (* s c) (* s c)))
     (/ 1.0 (* t_0 t_0)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x, c, 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, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s * x) * c
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-5d-109)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = (s * x) * c
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -5e-109:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = (s * x) * c;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[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], -5e-109], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$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))) < -5.0000000000000002e-109

   1. Initial program 75.1%

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

   5. Applied rewrites 57.4%

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

   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 57.7%

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

   10. Applied rewrites 57.7%

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

   12. Applied rewrites 57.7%

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

   if -5.0000000000000002e-109 < (/.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 64.3%

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 88.2%

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 83.5%

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

Alternative 3: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot x\right)\right)\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -5e-109)
   (/ -2.0 (* (* s c) (* s c)))
   (/ 1.0 (* (* c (* (* c x) (* s x))) s))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / ((c * ((c * x) * (s * x))) * s);
	}
	return tmp;
}

Fortran

NOTE: x, c, 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, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-5d-109)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / ((c * ((c * x) * (s * x))) * s)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / ((c * ((c * x) * (s * x))) * s);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -5e-109:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / ((c * ((c * x) * (s * x))) * s)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(Float64(c * Float64(Float64(c * x) * Float64(s * x))) * s));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / ((c * ((c * x) * (s * x))) * s);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[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], -5e-109], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(N[(c * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $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))) < -5.0000000000000002e-109

   1. Initial program 75.1%

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

   5. Applied rewrites 57.4%

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

   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 57.7%

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

   10. Applied rewrites 57.7%

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

   12. Applied rewrites 57.7%

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

   if -5.0000000000000002e-109 < (/.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 64.3%

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.0%

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

Alternative 4: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot s\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -5e-109)
   (/ -2.0 (* (* s c) (* s c)))
   (/ 1.0 (* (* (* (* (* c c) x) x) s) s))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	}
	return tmp;
}

Fortran

NOTE: x, c, 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, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-5d-109)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / (((((c * c) * x) * x) * s) * s)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -5e-109:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c * c) * x) * x) * s) * s));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[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], -5e-109], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * s), $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))) < -5.0000000000000002e-109

   1. Initial program 75.1%

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

   5. Applied rewrites 57.4%

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

   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 57.7%

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

   10. Applied rewrites 57.7%

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

   12. Applied rewrites 57.7%

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

   if -5.0000000000000002e-109 < (/.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 64.3%

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.1%

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

Alternative 5: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -5e-109)
   (/ -2.0 (* (* s c) (* s c)))
   (/ 1.0 (* (* (* (* x x) (* c c)) s) s))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / ((((x * x) * (c * c)) * s) * s);
	}
	return tmp;
}

Fortran

NOTE: x, c, 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, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-5d-109)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / ((((x * x) * (c * c)) * s) * s)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -5e-109) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / ((((x * x) * (c * c)) * s) * s);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -5e-109:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / ((((x * x) * (c * c)) * s) * s)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(x * x) * Float64(c * c)) * s) * s));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -5e-109)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / ((((x * x) * (c * c)) * s) * s);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[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], -5e-109], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * s), $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))) < -5.0000000000000002e-109

   1. Initial program 75.1%

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

   5. Applied rewrites 57.4%

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

   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 57.7%

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

   10. Applied rewrites 57.7%

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

   12. Applied rewrites 57.7%

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

   if -5.0000000000000002e-109 < (/.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 64.3%

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.0%

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

Alternative 6: 74.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ \mathbf{if}\;x \leq 0.00011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot x\right)\right)\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)))
   (if (<= x 0.00011)
     (/ (fma (fma 0.6666666666666666 (* x x) -2.0) (* x x) 1.0) (* t_0 t_0))
     (/ (cos (+ x x)) (* (* c (* (* c x) (* s x))) s)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if (x <= 0.00011) {
		tmp = fma(fma(0.6666666666666666, (x * x), -2.0), (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / ((c * ((c * x) * (s * x))) * s);
	}
	return tmp;
}

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	tmp = 0.0
	if (x <= 0.00011)
		tmp = Float64(fma(fma(0.6666666666666666, Float64(x * x), -2.0), Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(c * Float64(Float64(c * x) * Float64(s * x))) * s));
	end
	return tmp
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[x, 0.00011], N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(c * N[(N[(c * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000004e-4

   1. Initial program 64.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 97.5%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 71.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right), x \cdot x, 1\right)}}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)} \]

   if 1.10000000000000004e-4 < x

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 87.6

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

   9. Applied rewrites 87.6%

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

Alternative 7: 28.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* (* (* s s) c) c)))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / (((s * s) * c) * c);
}

Fortran

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

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / (((s * s) * c) * c);
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / (((s * s) * c) * c)

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / (((s * s) * c) * c);
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

5. Applied rewrites 48.9%

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

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 28.1%

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

10. Applied rewrites 28.3%

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

Alternative 8: 28.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* (* s s) (* c c))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / ((s * s) * (c * c));
}

Fortran

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

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / ((s * s) * (c * c));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / ((s * s) * (c * c))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(Float64(s * s) * Float64(c * c)))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / ((s * s) * (c * c));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

5. Applied rewrites 48.9%

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

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 28.1%

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

10. Applied rewrites 28.3%

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

12. Applied rewrites 29.0%

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

Alternative 9: 26.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* (* s c) (* s c))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / ((s * c) * (s * c));
}

Fortran

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

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / ((s * c) * (s * c));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / ((s * c) * (s * c))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / ((s * c) * (s * c));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

5. Applied rewrites 48.9%

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

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 28.1%

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

10. Applied rewrites 28.3%

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

12. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2025026 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))