mixedcos

Percentage Accurate: 67.5% → 98.9%

Time: 11.1s

Alternatives: 9

Speedup: 24.1×

• 32.0% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x_m \cdot 2\right)\\ \mathbf{if}\;x_m \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{{\left(c_m \cdot \left(x_m \cdot s_m\right)\right)}^{2}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (cos (* x_m 2.0))))
   (if (<= x_m 3e+20)
     (* (/ 1.0 (pow (* c_m (* x_m s_m)) 2.0)) t_0)
     (/ t_0 (* s_m (* (* x_m c_m) (* s_m (* x_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((x_m * 2.0));
	double tmp;
	if (x_m <= 3e+20) {
		tmp = (1.0 / pow((c_m * (x_m * s_m)), 2.0)) * t_0;
	} else {
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    if (x_m <= 3d+20) then
        tmp = (1.0d0 / ((c_m * (x_m * s_m)) ** 2.0d0)) * t_0
    else
        tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 t_0 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 3e+20) {
		tmp = (1.0 / Math.pow((c_m * (x_m * s_m)), 2.0)) * t_0;
	} else {
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_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((x_m * 2.0))
	tmp = 0
	if x_m <= 3e+20:
		tmp = (1.0 / math.pow((c_m * (x_m * s_m)), 2.0)) * t_0
	else:
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_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 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 3e+20)
		tmp = Float64(Float64(1.0 / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0)) * t_0);
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_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((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 3e+20)
		tmp = (1.0 / ((c_m * (x_m * s_m)) ^ 2.0)) * t_0;
	else
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 3e+20], N[(N[(1.0 / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e20

   1. Initial program 63.4%

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

      1. clear-num 63.4%

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

      2. associate-/r/ 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-*r* 56.9%

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

      5. unpow2 56.9%

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

      6. *-commutative 56.9%

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

      7. associate-*r* 56.0%

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

      8. associate-*r* 56.9%

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

      9. pow-prod-down 76.5%

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

      10. pow-prod-down 96.9%

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

      11. *-commutative 96.9%

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

   3. Applied egg-rr 96.9%

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

   if 3e20 < x

   1. Initial program 58.2%

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

      1. add-cbrt-cube 56.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      2. pow3 56.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

      3. *-commutative 56.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

      4. associate-*r* 46.6%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

      5. unpow2 46.6%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

      6. *-commutative 46.6%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)}^{3}}} \]

      7. pow-prod-down 67.7%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)}^{3}}} \]

      8. pow-prod-down 79.6%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\color{blue}{\left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)}}^{3}}} \]

      9. *-commutative 79.6%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({\left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)}^{2}\right)}^{3}}} \]

   3. Applied egg-rr 79.6%

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

      1. rem-cbrt-cube 96.3%

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

      2. unpow2 96.3%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-*r* 94.3%

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

      7. *-commutative 94.3%

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

      8. associate-*l* 94.3%

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

   5. Applied egg-rr 94.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 2: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{{\left(c_m \cdot \left(x_m \cdot s_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot 2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 9.5e-6)
   (/ (cos (* x_m -2.0)) (pow (* c_m (* x_m s_m)) 2.0))
   (/ (cos (* x_m 2.0)) (* s_m (* (* x_m c_m) (* s_m (* x_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 <= 9.5e-6) {
		tmp = cos((x_m * -2.0)) / pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 9.5d-6) then
        tmp = cos((x_m * (-2.0d0))) / ((c_m * (x_m * s_m)) ** 2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 (x_m <= 9.5e-6) {
		tmp = Math.cos((x_m * -2.0)) / Math.pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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):
	tmp = 0
	if x_m <= 9.5e-6:
		tmp = math.cos((x_m * -2.0)) / math.pow((c_m * (x_m * s_m)), 2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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 <= 9.5e-6)
		tmp = Float64(cos(Float64(x_m * -2.0)) / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_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)
	tmp = 0.0;
	if (x_m <= 9.5e-6)
		tmp = cos((x_m * -2.0)) / ((c_m * (x_m * s_m)) ^ 2.0);
	else
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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_] := If[LessEqual[x$95$m, 9.5e-6], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000005e-6

   1. Initial program 62.3%

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

      1. associate-/r* 61.9%

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

      2. unpow2 61.9%

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

      3. sqr-neg 61.9%

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

      4. unpow2 61.9%

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

      5. associate-/r* 62.3%

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

      6. cos-neg 62.3%

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

      7. *-commutative 62.3%

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

      8. distribute-rgt-neg-in 62.3%

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

      9. metadata-eval 62.3%

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

      10. associate-*r* 62.5%

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

      11. *-commutative 62.5%

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

      12. unpow2 62.5%

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

      13. sqr-neg 62.5%

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

      14. associate-*l* 72.2%

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

      15. associate-*r* 73.3%

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

      16. associate-*r* 71.6%

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

      17. associate-*r* 65.2%

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

      18. unpow2 65.2%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in x around inf 55.6%

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

      1. associate-/r* 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

      4. unpow2 55.6%

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

      5. swap-sqr 75.1%

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

      6. unpow2 75.1%

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

      7. associate-/r* 75.8%

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

      8. *-commutative 75.8%

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

      9. unpow2 75.8%

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

      10. unpow2 75.8%

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

      11. swap-sqr 96.8%

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

      12. unpow2 96.8%

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

      13. *-commutative 96.8%

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

   6. Simplified 96.8%

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

   if 9.5000000000000005e-6 < x

   1. Initial program 62.5%

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

      1. add-cbrt-cube 60.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      2. pow3 60.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

      3. *-commutative 60.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

      4. associate-*r* 52.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

      5. unpow2 52.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

      6. *-commutative 52.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)}^{3}}} \]

      7. pow-prod-down 71.0%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)}^{3}}} \]

      8. pow-prod-down 81.7%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\color{blue}{\left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)}}^{3}}} \]

      9. *-commutative 81.7%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({\left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)}^{2}\right)}^{3}}} \]

   3. Applied egg-rr 81.7%

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

      1. rem-cbrt-cube 96.6%

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

      2. unpow2 96.6%

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

      3. associate-*r* 93.5%

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

      4. associate-*r* 91.9%

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

      5. *-commutative 91.9%

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

      6. associate-*r* 94.8%

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

      7. *-commutative 94.8%

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

      8. associate-*l* 94.8%

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

   5. Applied egg-rr 94.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 95.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-1}{c_m}}{\left(c_m \cdot \left(x_m \cdot s_m\right)\right) \cdot \left(x_m \cdot \left(-s_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot 2\right)}{\left(x_m \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right) \cdot \left(c_m \cdot s_m\right)}\\ \end{array} \end{array} \]

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 1.35e-28)
   (/ (/ -1.0 c_m) (* (* c_m (* x_m s_m)) (* x_m (- s_m))))
   (/ (cos (* x_m 2.0)) (* (* x_m (* s_m (* x_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) {
	double tmp;
	if (x_m <= 1.35e-28) {
		tmp = (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m));
	} else {
		tmp = cos((x_m * 2.0)) / ((x_m * (s_m * (x_m * c_m))) * (c_m * s_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.35d-28) then
        tmp = ((-1.0d0) / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m))
    else
        tmp = cos((x_m * 2.0d0)) / ((x_m * (s_m * (x_m * c_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 (x_m <= 1.35e-28) {
		tmp = (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m));
	} else {
		tmp = Math.cos((x_m * 2.0)) / ((x_m * (s_m * (x_m * c_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 x_m <= 1.35e-28:
		tmp = (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m))
	else:
		tmp = math.cos((x_m * 2.0)) / ((x_m * (s_m * (x_m * c_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 <= 1.35e-28)
		tmp = Float64(Float64(-1.0 / c_m) / Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(x_m * Float64(-s_m))));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(Float64(x_m * Float64(s_m * Float64(x_m * c_m))) * Float64(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 (x_m <= 1.35e-28)
		tmp = (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m));
	else
		tmp = cos((x_m * 2.0)) / ((x_m * (s_m * (x_m * c_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[x$95$m, 1.35e-28], N[(N[(-1.0 / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * (-s$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3499999999999999e-28

   1. Initial program 62.1%

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

      1. add-sqr-sqrt 59.3%

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

      2. pow2 59.3%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in x around 0 85.7%

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

      1. unpow2 85.7%

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

      2. frac-2neg 85.7%

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

      3. metadata-eval 85.7%

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

      4. associate-/r* 85.7%

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

      5. frac-times 83.3%

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

      6. neg-mul-1 83.3%

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

      7. distribute-neg-frac 83.3%

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

      8. metadata-eval 83.3%

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

      9. distribute-rgt-neg-in 83.3%

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

      10. *-commutative 83.3%

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

      11. distribute-rgt-neg-in 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 1.3499999999999999e-28 < x

   1. Initial program 63.2%

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

      1. add-cbrt-cube 59.0%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      2. pow3 59.0%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

      3. *-commutative 59.0%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

      4. associate-*r* 51.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

      5. unpow2 51.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

      6. *-commutative 51.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)}^{3}}} \]

      7. pow-prod-down 67.8%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)}^{3}}} \]

      8. pow-prod-down 77.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\color{blue}{\left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)}}^{3}}} \]

      9. *-commutative 77.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({\left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)}^{2}\right)}^{3}}} \]

   3. Applied egg-rr 77.1%

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

      1. rem-cbrt-cube 97.0%

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

      2. unpow2 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*r* 97.1%

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

      5. associate-*l* 91.5%

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

      6. *-commutative 91.5%

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

      7. associate-*r* 95.7%

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

      8. *-commutative 95.7%

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

      9. associate-*r* 98.3%

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

      10. *-commutative 98.3%

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

      11. associate-*l* 96.9%

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

   5. Applied egg-rr 96.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-1}{c}}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 4: 98.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot 2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 3.3e-9)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (* x_m 2.0)) (* s_m (* (* x_m c_m) (* s_m (* x_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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 3.3e-9) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 3.3d-9) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 3.3e-9) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 3.3e-9:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 3.3e-9)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 3.3e-9)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.3e-9], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.30000000000000018e-9

   1. Initial program 62.5%

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

      1. add-sqr-sqrt 59.8%

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

      2. pow2 59.8%

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

   3. Applied egg-rr 84.0%

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

   4. Taylor expanded in x around 0 86.2%

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

      1. unpow2 86.2%

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

      2. frac-2neg 86.2%

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

      3. metadata-eval 86.2%

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

      4. frac-2neg 86.2%

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

      5. metadata-eval 86.2%

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

      6. frac-times 86.2%

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

      7. metadata-eval 86.2%

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

      8. distribute-rgt-neg-in 86.2%

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

      9. *-commutative 86.2%

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

      10. distribute-rgt-neg-in 86.2%

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

      11. distribute-rgt-neg-in 86.2%

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

      12. *-commutative 86.2%

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

      13. distribute-rgt-neg-in 86.2%

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

   6. Applied egg-rr 86.2%

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

   if 3.30000000000000018e-9 < x

   1. Initial program 62.0%

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

      1. add-cbrt-cube 58.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      2. pow3 58.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

      3. *-commutative 58.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

      4. associate-*r* 50.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

      5. unpow2 50.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

      6. *-commutative 50.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)}^{3}}} \]

      7. pow-prod-down 68.7%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)}^{3}}} \]

      8. pow-prod-down 79.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\color{blue}{\left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)}}^{3}}} \]

      9. *-commutative 79.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\sqrt[3]{{\left({\left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)}^{2}\right)}^{3}}} \]

   3. Applied egg-rr 79.1%

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

      1. rem-cbrt-cube 96.7%

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

      2. unpow2 96.7%

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

      3. associate-*r* 93.6%

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

      4. associate-*r* 92.1%

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

      5. *-commutative 92.1%

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

      6. associate-*r* 95.0%

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

      7. *-commutative 95.0%

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

      8. associate-*l* 94.9%

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

   5. Applied egg-rr 94.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 5: 78.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{-1}{c_m}}{\left(c_m \cdot \left(x_m \cdot s_m\right)\right) \cdot \left(x_m \cdot \left(-s_m\right)\right)} \end{array} \]

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
 (/ (/ -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) {
	return (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_m * -s_m));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = ((-1.0d0) / c_m) / ((c_m * (x_m * s_m)) * (x_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 (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_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 (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_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(Float64(-1.0 / c_m) / Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(x_m * Float64(-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 = (-1.0 / c_m) / ((c_m * (x_m * s_m)) * (x_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[(N[(-1.0 / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * (-s$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

3. Applied egg-rr 77.6%

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

4. Taylor expanded in x around 0 82.0%

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

   1. unpow2 82.0%

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

   2. frac-2neg 82.0%

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

   3. metadata-eval 82.0%

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

   4. associate-/r* 82.0%

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

   5. frac-times 80.1%

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

   6. neg-mul-1 80.1%

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

   7. distribute-neg-frac 80.1%

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

   8. metadata-eval 80.1%

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

   9. distribute-rgt-neg-in 80.1%

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

   10. *-commutative 80.1%

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

   11. distribute-rgt-neg-in 80.1%

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

6. Applied egg-rr 80.1%

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

7. Final simplification 80.1%

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

Alternative 6: 77.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{1}{c_m \cdot \left(x_m \cdot s_m\right)}}{s_m \cdot \left(x_m \cdot c_m\right)} \end{array} \]

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
 (/ (/ 1.0 (* c_m (* x_m s_m))) (* s_m (* x_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 (1.0 / (c_m * (x_m * s_m))) / (s_m * (x_m * c_m));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (x_m * s_m))) / (s_m * (x_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 (1.0 / (c_m * (x_m * s_m))) / (s_m * (x_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 (1.0 / (c_m * (x_m * s_m))) / (s_m * (x_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(Float64(1.0 / Float64(c_m * Float64(x_m * s_m))) / Float64(s_m * Float64(x_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 = (1.0 / (c_m * (x_m * s_m))) / (s_m * (x_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[(N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

3. Applied egg-rr 77.6%

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

4. Taylor expanded in x around 0 82.0%

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

   1. unpow2 82.0%

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

   2. un-div-inv 82.0%

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

   3. *-commutative 82.0%

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

   4. associate-*r* 79.8%

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

   5. *-commutative 79.8%

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

   6. associate-*l* 81.4%

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

   7. *-commutative 81.4%

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

   8. associate-*r* 80.2%

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

   9. *-commutative 80.2%

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

   10. associate-*l* 82.4%

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

6. Applied egg-rr 82.4%

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

7. Taylor expanded in s around 0 81.4%

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

8. Final simplification 81.4%

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

Alternative 7: 77.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{1}{s_m \cdot \left(x_m \cdot c_m\right)}}{c_m \cdot \left(x_m \cdot s_m\right)} \end{array} \]

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
 (/ (/ 1.0 (* s_m (* x_m c_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) {
	return (1.0 / (s_m * (x_m * c_m))) / (c_m * (x_m * s_m));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (s_m * (x_m * c_m))) / (c_m * (x_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 (1.0 / (s_m * (x_m * c_m))) / (c_m * (x_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 (1.0 / (s_m * (x_m * c_m))) / (c_m * (x_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(Float64(1.0 / Float64(s_m * Float64(x_m * c_m))) / Float64(c_m * Float64(x_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 = (1.0 / (s_m * (x_m * c_m))) / (c_m * (x_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[(N[(1.0 / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

3. Applied egg-rr 77.6%

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

4. Taylor expanded in x around 0 82.0%

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

   1. unpow2 82.0%

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

   2. un-div-inv 82.0%

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

   3. *-commutative 82.0%

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

   4. associate-*r* 79.8%

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

   5. *-commutative 79.8%

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

   6. associate-*l* 81.4%

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

   7. *-commutative 81.4%

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

   8. associate-*r* 80.2%

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

   9. *-commutative 80.2%

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

   10. associate-*l* 82.4%

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

6. Applied egg-rr 82.4%

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

7. Taylor expanded in s around 0 81.4%

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

8. Final simplification 81.4%

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

Alternative 8: 77.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s_m \cdot \left(x_m \cdot c_m\right)\\ \frac{\frac{1}{t_0}}{t_0} \end{array} \end{array} \]

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 (* s_m (* x_m c_m)))) (/ (/ 1.0 t_0) t_0)))

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 = s_m * (x_m * c_m);
	return (1.0 / t_0) / t_0;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (x_m * c_m)
    code = (1.0d0 / t_0) / t_0
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 t_0 = s_m * (x_m * c_m);
	return (1.0 / t_0) / t_0;
}

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 = s_m * (x_m * c_m)
	return (1.0 / t_0) / t_0

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(s_m * Float64(x_m * c_m))
	return Float64(Float64(1.0 / t_0) / t_0)
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)
	t_0 = s_m * (x_m * c_m);
	tmp = (1.0 / t_0) / t_0;
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[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

3. Applied egg-rr 77.6%

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

4. Taylor expanded in x around 0 82.0%

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

   1. unpow2 82.0%

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

   2. un-div-inv 82.0%

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

   3. *-commutative 82.0%

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

   4. associate-*r* 79.8%

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

   5. *-commutative 79.8%

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

   6. associate-*l* 81.4%

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

   7. *-commutative 81.4%

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

   8. associate-*r* 80.2%

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

   9. *-commutative 80.2%

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

   10. associate-*l* 82.4%

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

6. Applied egg-rr 82.4%

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

7. Final simplification 82.4%

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

Alternative 9: 79.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \end{array} \]

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 (* c_m (* x_m s_m)))) (/ 1.0 (* t_0 t_0))))

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 = c_m * (x_m * s_m);
	return 1.0 / (t_0 * t_0);
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x_m * s_m)
    code = 1.0d0 / (t_0 * t_0)
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 t_0 = c_m * (x_m * s_m);
	return 1.0 / (t_0 * t_0);
}

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 = c_m * (x_m * s_m)
	return 1.0 / (t_0 * t_0)

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(c_m * Float64(x_m * s_m))
	return Float64(1.0 / Float64(t_0 * t_0))
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)
	t_0 = c_m * (x_m * s_m);
	tmp = 1.0 / (t_0 * t_0);
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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

3. Applied egg-rr 77.6%

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

4. Taylor expanded in x around 0 82.0%

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

   1. unpow2 82.0%

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

   2. frac-2neg 82.0%

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

   3. metadata-eval 82.0%

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

   4. frac-2neg 82.0%

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

   5. metadata-eval 82.0%

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

   6. frac-times 82.0%

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

   7. metadata-eval 82.0%

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

   8. distribute-rgt-neg-in 82.0%

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

   9. *-commutative 82.0%

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

   10. distribute-rgt-neg-in 82.0%

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

   11. distribute-rgt-neg-in 82.0%

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

   12. *-commutative 82.0%

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

   13. distribute-rgt-neg-in 82.0%

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

6. Applied egg-rr 82.0%

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

7. Final simplification 82.0%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))