mixedcos

Percentage Accurate: 66.0% → 96.8%

Time: 6.5s

Alternatives: 15

Speedup: 2.3×

• 32.1% 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: 66.0% 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: 96.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.6e-99)
   (/ (fma (* x_m x_m) -2.0 1.0) (* (* (pow (* s_m x_m) 2.0) c_m) c_m))
   (/ (cos (* 2.0 x_m)) (* (* (* (* c_m x_m) s_m) (* c_m x_m)) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e-99) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	} else {
		tmp = cos((2.0 * x_m)) / ((((c_m * x_m) * s_m) * (c_m * x_m)) * s_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.6e-99)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * Float64(c_m * x_m)) * s_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.6e-99], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000001e-99

   1. Initial program 62.2%

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

   3. 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 47.8%

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

   7. Applied rewrites 56.4%

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

   if 3.6000000000000001e-99 < x

   1. Initial program 69.7%

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 88.3%

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 92.5%

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

Alternative 2: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{1}{s\_m \cdot \left(c\_m \cdot c\_m\right)}}{x\_m}}{s\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(x\_m \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m}}{s\_m \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0
         (/
          (cos (* 2.0 x_m))
          (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))))
   (if (<= t_0 -5e-161)
     (/ -2.0 (* c_m (* s_m (* s_m c_m))))
     (if (<= t_0 INFINITY)
       (/ (/ (/ 1.0 (* s_m (* c_m c_m))) x_m) (* s_m x_m))
       (/ (/ 1.0 (* (* (* x_m x_m) s_m) c_m)) (* s_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((1.0 / (s_m * (c_m * c_m))) / x_m) / (s_m * x_m);
	} else {
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((1.0 / (s_m * (c_m * c_m))) / x_m) / (s_m * x_m);
	} else {
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))
	tmp = 0
	if t_0 <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	elif t_0 <= math.inf:
		tmp = ((1.0 / (s_m * (c_m * c_m))) / x_m) / (s_m * x_m)
	else:
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m)))
	tmp = 0.0
	if (t_0 <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(1.0 / Float64(s_m * Float64(c_m * c_m))) / x_m) / Float64(s_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(x_m * x_m) * s_m) * c_m)) / Float64(s_m * c_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m));
	tmp = 0.0;
	if (t_0 <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	elseif (t_0 <= Inf)
		tmp = ((1.0 / (s_m * (c_m * c_m))) / x_m) / (s_m * x_m);
	else
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(1.0 / N[(s$95$m * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * c$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 81.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

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

   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. 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 13.8%

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

   7. Applied rewrites 66.9%

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 38.6%

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

Alternative 3: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{1}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot x\_m}}{s\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(x\_m \cdot x\_m\right) \cdot s\_m\right) \cdot c\_m}}{s\_m \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0
         (/
          (cos (* 2.0 x_m))
          (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))))
   (if (<= t_0 -5e-161)
     (/ -2.0 (* c_m (* s_m (* s_m c_m))))
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 (* (* (* c_m c_m) s_m) x_m)) (* s_m x_m))
       (/ (/ 1.0 (* (* (* x_m x_m) s_m) c_m)) (* s_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	} else {
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	} else {
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))
	tmp = 0
	if t_0 <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	elif t_0 <= math.inf:
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m)
	else:
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m)))
	tmp = 0.0
	if (t_0 <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(c_m * c_m) * s_m) * x_m)) / Float64(s_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(x_m * x_m) * s_m) * c_m)) / Float64(s_m * c_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m));
	tmp = 0.0;
	if (t_0 <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	elseif (t_0 <= Inf)
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	else
		tmp = (1.0 / (((x_m * x_m) * s_m) * c_m)) / (s_m * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * c$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 81.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 84.0%

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

   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. 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 13.8%

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

   7. Applied rewrites 66.9%

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 38.6%

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

Alternative 4: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{1}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot x\_m}}{s\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0
         (/
          (cos (* 2.0 x_m))
          (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))))
   (if (<= t_0 -5e-161)
     (/ -2.0 (* c_m (* s_m (* s_m c_m))))
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 (* (* (* c_m c_m) s_m) x_m)) (* s_m x_m))
       (/ 1.0 (* (* (* (* (* c_m x_m) s_m) x_m) c_m) s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	} else {
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m));
	double tmp;
	if (t_0 <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	} else {
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))
	tmp = 0
	if t_0 <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	elif t_0 <= math.inf:
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m)
	else:
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m)))
	tmp = 0.0
	if (t_0 <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(c_m * c_m) * s_m) * x_m)) / Float64(s_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * x_m) * c_m) * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m));
	tmp = 0.0;
	if (t_0 <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	elseif (t_0 <= Inf)
		tmp = (1.0 / (((c_m * c_m) * s_m) * x_m)) / (s_m * x_m);
	else
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 81.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 84.0%

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

   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. 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 13.8%

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

   7. Applied rewrites 66.9%

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 57.3%

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

Alternative 5: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(s\_m \cdot c\_m\right) \cdot c\_m}}{x\_m}}{s\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -5e-161)
   (/ (fma (* x_m x_m) -2.0 1.0) (* (* (pow (* s_m x_m) 2.0) c_m) c_m))
   (/ (/ (/ 1.0 (* (* s_m c_m) c_m)) x_m) (* s_m x_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	} else {
		tmp = ((1.0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(s_m * c_m) * c_m)) / x_m) / Float64(s_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-161], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(s$95$m * x$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   7. Applied rewrites 38.4%

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

   if -4.9999999999999999e-161 < (/.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 64.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.8%

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

   10. Applied rewrites 76.1%

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

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(s\_m \cdot c\_m\right) \cdot c\_m}}{x\_m}}{s\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -5e-161)
   (/ -2.0 (* c_m (* s_m (* s_m c_m))))
   (/ (/ (/ 1.0 (* (* s_m c_m) c_m)) x_m) (* s_m x_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = ((1.0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m);
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-5d-161)) then
        tmp = (-2.0d0) / (c_m * (s_m * (s_m * c_m)))
    else
        tmp = ((1.0d0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = ((1.0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	else:
		tmp = ((1.0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(s_m * c_m) * c_m)) / x_m) / Float64(s_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	else
		tmp = ((1.0 / ((s_m * c_m) * c_m)) / x_m) / (s_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(s$95$m * x$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 64.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.8%

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

   10. Applied rewrites 76.1%

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

Alternative 7: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -5e-161)
   (/ -2.0 (* c_m (* s_m (* s_m c_m))))
   (/ 1.0 (* (* (* (* (* c_m x_m) s_m) x_m) c_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-5d-161)) then
        tmp = (-2.0d0) / (c_m * (s_m * (s_m * c_m)))
    else
        tmp = 1.0d0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	else:
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * x_m) * c_m) * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	else
		tmp = 1.0 / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 64.6%

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

   3. 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 71.1%

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

   7. Applied rewrites 87.0%

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

   9. Applied rewrites 89.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 78.5%

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

Alternative 8: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot \left(\left(c\_m \cdot x\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right)\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -5e-161)
   (/ -2.0 (* c_m (* s_m (* s_m c_m))))
   (/ 1.0 (* (* c_m (* (* c_m x_m) (* s_m x_m))) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m);
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-5d-161)) then
        tmp = (-2.0d0) / (c_m * (s_m * (s_m * c_m)))
    else
        tmp = 1.0d0 / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	else:
		tmp = 1.0 / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * Float64(Float64(c_m * x_m) * Float64(s_m * x_m))) * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	else
		tmp = 1.0 / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * N[(N[(c$95$m * x$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * s$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 64.6%

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

   3. 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 71.1%

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

   7. Applied rewrites 87.0%

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

   8. 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 77.1%

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

Alternative 9: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot s\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -5e-161)
   (/ -2.0 (* c_m (* s_m (* s_m c_m))))
   (/ 1.0 (* (* (* (* (* c_m c_m) x_m) x_m) s_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	}
	return tmp;
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-5d-161)) then
        tmp = (-2.0d0) / (c_m * (s_m * (s_m * c_m)))
    else
        tmp = 1.0d0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -5e-161) {
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	} else {
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -5e-161:
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)))
	else:
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = Float64(-2.0 / Float64(c_m * Float64(s_m * Float64(s_m * c_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c_m * c_m) * x_m) * x_m) * s_m) * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -5e-161)
		tmp = -2.0 / (c_m * (s_m * (s_m * c_m)));
	else
		tmp = 1.0 / (((((c_m * c_m) * x_m) * x_m) * s_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-161], N[(-2.0 / N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $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))) < -4.9999999999999999e-161

   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

   5. Applied rewrites 37.5%

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

   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 38.5%

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

   10. Applied rewrites 38.5%

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

   12. Applied rewrites 38.5%

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

   if -4.9999999999999999e-161 < (/.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 64.6%

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

   3. 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 71.1%

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

   6. 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 65.6%

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

Alternative 10: 95.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3e-126)
   (/ (fma (* x_m x_m) -2.0 1.0) (* (* (pow (* s_m x_m) 2.0) c_m) c_m))
   (/ (cos (* 2.0 x_m)) (* (* (* (* c_m x_m) s_m) x_m) (* s_m c_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3e-126) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	} else {
		tmp = cos((2.0 * x_m)) / ((((c_m * x_m) * s_m) * x_m) * (s_m * c_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3e-126)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * x_m) * Float64(s_m * c_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3e-126], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000002e-126

   1. Initial program 63.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 47.3%

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

   7. Applied rewrites 55.7%

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

   if 3.0000000000000002e-126 < x

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 87.4%

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 90.4%

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

Alternative 11: 94.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.6e-99)
   (/ (fma (* x_m x_m) -2.0 1.0) (* (* (pow (* s_m x_m) 2.0) c_m) c_m))
   (/ (cos (+ x_m x_m)) (* (* (* (* (* c_m x_m) s_m) x_m) c_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e-99) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	} else {
		tmp = cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.6e-99)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * x_m) * c_m) * s_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.6e-99], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000001e-99

   1. Initial program 62.2%

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

   3. 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 47.8%

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

   7. Applied rewrites 56.4%

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

   if 3.6000000000000001e-99 < x

   1. Initial program 69.7%

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 88.3%

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

   9. Applied rewrites 89.4%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 89.4

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

   11. Applied rewrites 89.4%

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

Alternative 12: 93.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, -2, 1\right)}{\left({\left(s\_m \cdot x\_m\right)}^{2} \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot \left(\left(c\_m \cdot x\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right)\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.6e-99)
   (/ (fma (* x_m x_m) -2.0 1.0) (* (* (pow (* s_m x_m) 2.0) c_m) c_m))
   (/ (cos (+ x_m x_m)) (* (* c_m (* (* c_m x_m) (* s_m x_m))) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e-99) {
		tmp = fma((x_m * x_m), -2.0, 1.0) / ((pow((s_m * x_m), 2.0) * c_m) * c_m);
	} else {
		tmp = cos((x_m + x_m)) / ((c_m * ((c_m * x_m) * (s_m * x_m))) * s_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.6e-99)
		tmp = Float64(fma(Float64(x_m * x_m), -2.0, 1.0) / Float64(Float64((Float64(s_m * x_m) ^ 2.0) * c_m) * c_m));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * Float64(Float64(c_m * x_m) * Float64(s_m * x_m))) * s_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.6e-99], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * N[(N[(c$95$m * x$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000001e-99

   1. Initial program 62.2%

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

   3. 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 47.8%

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

   7. Applied rewrites 56.4%

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

   if 3.6000000000000001e-99 < x

   1. Initial program 69.7%

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 88.3%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(\left(c \cdot x\right) \cdot \left(s \cdot x\right)\right)\right) \cdot s} \]
   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. lift-+.f64 88.3

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

   9. Applied rewrites 88.3%

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

Alternative 13: 30.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * ((s_m * s_m) * c_m));
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (-2.0d0) / (c_m * ((s_m * s_m) * c_m))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * ((s_m * s_m) * c_m));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (c_m * ((s_m * s_m) * c_m))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

5. Applied rewrites 44.2%

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

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 27.8%

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

10. Applied rewrites 27.8%

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

Alternative 14: 27.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * (s_m * (s_m * c_m)));
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (-2.0d0) / (c_m * (s_m * (s_m * c_m)))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * (s_m * (s_m * c_m)));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (c_m * (s_m * (s_m * c_m)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

5. Applied rewrites 44.2%

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

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 27.8%

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

10. Applied rewrites 27.8%

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

12. Applied rewrites 25.0%

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

Alternative 15: 25.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((s_m * c_m) * c_m) * s_m);
}

Fortran

x_m =     private
c_m =     private
s_m =     private
NOTE: x_m, c_m, and s_m 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_m, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (-2.0d0) / (((s_m * c_m) * c_m) * s_m)
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((s_m * c_m) * c_m) * s_m);
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (((s_m * c_m) * c_m) * s_m)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

5. Applied rewrites 44.2%

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

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 27.8%

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

10. Applied rewrites 27.8%

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

11. Taylor expanded in c around 0

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

13. Applied rewrites 23.7%

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

Reproduce

herbie shell --seed 2025056 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))