mixedcos

Percentage Accurate: 66.7% → 98.5%

Time: 10.4s

Alternatives: 9

Speedup: 24.1×

• 31.7% 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: 66.7% 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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(s \cdot \sqrt{x}\right)\\ \frac{\cos \left(2 \cdot x\right)}{t_0 \cdot \left(x \cdot t_0\right)} \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s (sqrt x))))) (/ (cos (* 2.0 x)) (* t_0 (* x t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (s * sqrt(x));
	return cos((2.0 * x)) / (t_0 * (x * t_0));
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (s * sqrt(x))
    code = cos((2.0d0 * x)) / (t_0 * (x * t_0))
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (s * Math.sqrt(x));
	return Math.cos((2.0 * x)) / (t_0 * (x * t_0));
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (s * math.sqrt(x))
	return math.cos((2.0 * x)) / (t_0 * (x * t_0))

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(s * sqrt(x)))
	return Float64(cos(Float64(2.0 * x)) / Float64(t_0 * Float64(x * t_0)))
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (s * sqrt(x));
	tmp = cos((2.0 * x)) / (t_0 * (x * t_0));
end

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 66.3%

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

   1. unpow2 66.3%

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

   2. unpow2 66.3%

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

   3. unpow2 66.3%

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

   4. swap-sqr 79.9%

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

   5. swap-sqr 97.0%

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

   6. unpow2 97.0%

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

6. Simplified 97.0%

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

   1. associate-*r* 97.8%

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

   2. unpow-prod-down 80.1%

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

   3. pow-prod-down 66.2%

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

   4. pow2 66.2%

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

   5. pow2 66.2%

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

   6. associate-*r* 70.3%

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

   7. pow2 70.3%

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

   8. *-commutative 70.3%

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

   9. associate-*r* 76.8%

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

   10. add-sqr-sqrt 39.1%

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

   11. associate-*r* 39.1%

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

8. Applied egg-rr 50.3%

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

9. Final simplification 50.3%

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

Alternative 2: 92.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;s \leq 1.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= s 1.5e+173)
     (/ (cos (* 2.0 x)) (* c (* x (* c (* s (* x s))))))
     (/ 1.0 (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (s <= 1.5e+173) {
		tmp = cos((2.0 * x)) / (c * (x * (c * (s * (x * s)))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (s <= 1.5d+173) then
        tmp = cos((2.0d0 * x)) / (c * (x * (c * (s * (x * s)))))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (s <= 1.5e+173) {
		tmp = Math.cos((2.0 * x)) / (c * (x * (c * (s * (x * s)))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if s <= 1.5e+173:
		tmp = math.cos((2.0 * x)) / (c * (x * (c * (s * (x * s)))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (s <= 1.5e+173)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(c * Float64(x * Float64(c * Float64(s * Float64(x * s))))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (s <= 1.5e+173)
		tmp = cos((2.0 * x)) / (c * (x * (c * (s * (x * s)))));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 1.5e+173], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(x * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.4999999999999999e173

   1. Initial program 71.3%

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

      1. associate-/r* 71.3%

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

      2. remove-double-neg 71.3%

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

      3. distribute-lft-neg-out 71.3%

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

      4. distribute-lft-neg-out 71.3%

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

      5. distribute-rgt-neg-out 71.3%

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

      6. associate-/l/ 71.3%

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

      7. distribute-rgt-neg-out 71.3%

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

      8. distribute-lft-neg-out 71.3%

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

      9. associate-*l* 75.3%

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

      10. distribute-lft-neg-in 75.3%

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

      11. distribute-lft-neg-out 75.3%

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

      12. remove-double-neg 75.3%

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

      13. associate-*r* 75.4%

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

      14. *-commutative 75.4%

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

      15. associate-*r* 72.3%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in x around 0 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. unpow2 66.8%

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

      4. swap-sqr 79.4%

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

      5. swap-sqr 97.1%

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

      6. unpow2 97.1%

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

   6. Simplified 97.1%

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

      1. unpow2 97.1%

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

      2. swap-sqr 79.4%

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

      3. associate-*r* 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*l* 71.3%

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

      6. associate-*l* 75.0%

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

      7. associate-*l* 78.8%

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

      8. associate-*l* 80.7%

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

      9. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

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

   if 1.4999999999999999e173 < s

   1. Initial program 70.1%

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

      1. associate-/r* 70.1%

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

      2. remove-double-neg 70.1%

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

      3. distribute-lft-neg-out 70.1%

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

      4. distribute-lft-neg-out 70.1%

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

      5. distribute-rgt-neg-out 70.1%

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

      6. associate-/l/ 70.1%

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

      7. distribute-rgt-neg-out 70.1%

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

      8. distribute-lft-neg-out 70.1%

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

      9. associate-*l* 70.1%

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

      10. distribute-lft-neg-in 70.1%

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

      11. distribute-lft-neg-out 70.1%

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

      12. remove-double-neg 70.1%

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

      13. associate-*r* 70.1%

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

      14. *-commutative 70.1%

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

      15. associate-*r* 70.1%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around 0 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. unpow2 61.9%

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

      4. swap-sqr 84.9%

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

      5. swap-sqr 96.0%

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

      6. unpow2 96.0%

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

   6. Simplified 96.0%

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

      1. unpow2 96.0%

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

   8. Applied egg-rr 96.0%

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

   9. Taylor expanded in x around 0 92.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]

Alternative 3: 92.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;s \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t_0}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (if (<= s 5e-17)
     (/ t_0 (* c (* x (* c (* s (* x s))))))
     (/ t_0 (* (* c s) (* x (* c (* x s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (s <= 5e-17) {
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	} else {
		tmp = t_0 / ((c * s) * (x * (c * (x * s))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if (s <= 5d-17) then
        tmp = t_0 / (c * (x * (c * (s * (x * s)))))
    else
        tmp = t_0 / ((c * s) * (x * (c * (x * s))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((2.0 * x));
	double tmp;
	if (s <= 5e-17) {
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	} else {
		tmp = t_0 / ((c * s) * (x * (c * (x * s))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if s <= 5e-17:
		tmp = t_0 / (c * (x * (c * (s * (x * s)))))
	else:
		tmp = t_0 / ((c * s) * (x * (c * (x * s))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (s <= 5e-17)
		tmp = Float64(t_0 / Float64(c * Float64(x * Float64(c * Float64(s * Float64(x * s))))));
	else
		tmp = Float64(t_0 / Float64(Float64(c * s) * Float64(x * Float64(c * Float64(x * s)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (s <= 5e-17)
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	else
		tmp = t_0 / ((c * s) * (x * (c * (x * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 5e-17], N[(t$95$0 / N[(c * N[(x * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 4.9999999999999999e-17

   1. Initial program 71.1%

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

      1. associate-/r* 71.1%

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

      2. remove-double-neg 71.1%

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

      3. distribute-lft-neg-out 71.1%

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

      4. distribute-lft-neg-out 71.1%

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

      5. distribute-rgt-neg-out 71.1%

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

      6. associate-/l/ 71.1%

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

      7. distribute-rgt-neg-out 71.1%

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

      8. distribute-lft-neg-out 71.1%

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

      9. associate-*l* 73.5%

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

      10. distribute-lft-neg-in 73.5%

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

      11. distribute-lft-neg-out 73.5%

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

      12. remove-double-neg 73.5%

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

      13. associate-*r* 73.0%

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

      14. *-commutative 73.0%

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

      15. associate-*r* 71.3%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 65.8%

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

      1. unpow2 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

      4. swap-sqr 80.7%

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

      5. swap-sqr 97.2%

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

      6. unpow2 97.2%

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

   6. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. swap-sqr 80.7%

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

      3. associate-*r* 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*l* 71.1%

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

      6. associate-*l* 73.0%

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

      7. associate-*l* 77.5%

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

      8. associate-*l* 79.2%

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

      9. associate-*l* 87.2%

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

   8. Applied egg-rr 87.2%

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

   if 4.9999999999999999e-17 < s

   1. Initial program 71.4%

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

      1. associate-/r* 71.4%

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

      2. remove-double-neg 71.4%

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

      3. distribute-lft-neg-out 71.4%

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

      4. distribute-lft-neg-out 71.4%

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

      5. distribute-rgt-neg-out 71.4%

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

      6. associate-/l/ 71.4%

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

      7. distribute-rgt-neg-out 71.4%

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

      8. distribute-lft-neg-out 71.4%

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

      9. associate-*l* 79.1%

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

      10. distribute-lft-neg-in 79.1%

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

      11. distribute-lft-neg-out 79.1%

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

      12. remove-double-neg 79.1%

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

      13. associate-*r* 80.7%

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

      14. *-commutative 80.7%

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

      15. associate-*r* 74.5%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in x around 0 68.0%

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

      1. unpow2 68.0%

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

      2. unpow2 68.0%

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

      3. unpow2 68.0%

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

      4. swap-sqr 77.7%

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

      5. swap-sqr 96.4%

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

      6. unpow2 96.4%

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

   6. Simplified 96.4%

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

      1. unpow2 96.4%

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

      2. associate-*r* 95.0%

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

      3. associate-*l* 91.9%

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

   8. Applied egg-rr 91.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 4: 96.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\cos \left(2 \cdot x\right)}{t_0 \cdot t_0} \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (cos (* 2.0 x)) (* t_0 t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return cos((2.0 * x)) / (t_0 * t_0);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = cos((2.0d0 * x)) / (t_0 * t_0)
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return Math.cos((2.0 * x)) / (t_0 * t_0);
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return math.cos((2.0 * x)) / (t_0 * t_0)

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0))
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = cos((2.0 * x)) / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 66.3%

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

   1. unpow2 66.3%

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

   2. unpow2 66.3%

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

   3. unpow2 66.3%

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

   4. swap-sqr 79.9%

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

   5. swap-sqr 97.0%

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

   6. unpow2 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

8. Applied egg-rr 97.0%

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

9. Final simplification 97.0%

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

Alternative 5: 71.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4.2e-55)
   (/ 1.0 (* x (* (* s (* x s)) (* c c))))
   (/ 1.0 (* x (* x (* c (* c (* s s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-55) {
		tmp = 1.0 / (x * ((s * (x * s)) * (c * c)));
	} else {
		tmp = 1.0 / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4.2d-55) then
        tmp = 1.0d0 / (x * ((s * (x * s)) * (c * c)))
    else
        tmp = 1.0d0 / (x * (x * (c * (c * (s * s)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-55) {
		tmp = 1.0 / (x * ((s * (x * s)) * (c * c)));
	} else {
		tmp = 1.0 / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4.2e-55:
		tmp = 1.0 / (x * ((s * (x * s)) * (c * c)))
	else:
		tmp = 1.0 / (x * (x * (c * (c * (s * s)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4.2e-55)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(s * Float64(x * s)) * Float64(c * c))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4.2e-55)
		tmp = 1.0 / (x * ((s * (x * s)) * (c * c)));
	else
		tmp = 1.0 / (x * (x * (c * (c * (s * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4.2e-55], N[(1.0 / N[(x * N[(N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(x * N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000003e-55

   1. Initial program 71.6%

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

      1. associate-/r* 71.6%

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

      2. remove-double-neg 71.6%

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

      3. distribute-lft-neg-out 71.6%

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

      4. distribute-lft-neg-out 71.6%

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

      5. distribute-rgt-neg-out 71.6%

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

      6. associate-/l/ 71.6%

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

      7. distribute-rgt-neg-out 71.6%

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

      8. distribute-lft-neg-out 71.6%

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

      9. associate-*l* 75.2%

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

      10. distribute-lft-neg-in 75.2%

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

      11. distribute-lft-neg-out 75.2%

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

      12. remove-double-neg 75.2%

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

      13. associate-*r* 74.7%

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

      14. *-commutative 74.7%

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

      15. associate-*r* 71.7%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in x around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in s around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r* 82.0%

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

   9. Simplified 82.0%

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

   10. Taylor expanded in x around 0 75.0%

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

   if 4.2000000000000003e-55 < x

   1. Initial program 70.1%

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

      1. associate-/r* 70.1%

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

      2. remove-double-neg 70.1%

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

      3. distribute-lft-neg-out 70.1%

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

      4. distribute-lft-neg-out 70.1%

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

      5. distribute-rgt-neg-out 70.1%

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

      6. associate-/l/ 70.1%

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

      7. distribute-rgt-neg-out 70.1%

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

      8. distribute-lft-neg-out 70.1%

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

      9. associate-*l* 73.9%

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

      10. distribute-lft-neg-in 73.9%

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

      11. distribute-lft-neg-out 73.9%

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

      12. remove-double-neg 73.9%

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

      13. associate-*r* 75.3%

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

      14. *-commutative 75.3%

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

      15. associate-*r* 72.8%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in x around 0 60.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 64.2% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ 1.0 (* x (* x (* c (* c (* s s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 1.0 / (x * (x * (c * (c * (s * s)))));
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / (x * (x * (c * (c * (s * s)))))
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return 1.0 / (x * (x * (c * (c * (s * s)))));
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0 / (x * (x * (c * (c * (s * s)))))

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(1.0 / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))))
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 1.0 / (x * (x * (c * (c * (s * s)))));
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(1.0 / N[(x * N[(x * N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 67.5%

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

5. Final simplification 67.5%

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

Alternative 7: 76.8% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c s) (* x (* c (* x s))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 1.0 / ((c * s) * (x * (c * (x * s))));
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * s) * (x * (c * (x * s))))
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return 1.0 / ((c * s) * (x * (c * (x * s))));
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0 / ((c * s) * (x * (c * (x * s))))

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * s) * Float64(x * Float64(c * Float64(x * s)))))
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 1.0 / ((c * s) * (x * (c * (x * s))));
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 66.3%

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

   1. unpow2 66.3%

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

   2. unpow2 66.3%

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

   3. unpow2 66.3%

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

   4. swap-sqr 79.9%

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

   5. swap-sqr 97.0%

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

   6. unpow2 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

   2. associate-*r* 95.6%

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

   3. associate-*l* 93.0%

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

8. Applied egg-rr 93.0%

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

9. Taylor expanded in x around 0 78.5%

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

10. Final simplification 78.5%

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

Alternative 8: 79.9% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 66.3%

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

   1. unpow2 66.3%

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

   2. unpow2 66.3%

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

   3. unpow2 66.3%

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

   4. swap-sqr 79.9%

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

   5. swap-sqr 97.0%

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

   6. unpow2 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

8. Applied egg-rr 97.0%

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

9. Taylor expanded in x around 0 79.5%

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

10. Final simplification 79.5%

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

Alternative 9: 27.7% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = (-2.0d0) / ((c * c) * (s * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 71.1%

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

   2. remove-double-neg 71.1%

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

   3. distribute-lft-neg-out 71.1%

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

   4. distribute-lft-neg-out 71.1%

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

   5. distribute-rgt-neg-out 71.1%

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

   6. associate-/l/ 71.1%

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

   7. distribute-rgt-neg-out 71.1%

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

   8. distribute-lft-neg-out 71.1%

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

   9. associate-*l* 74.8%

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

   10. distribute-lft-neg-in 74.8%

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

   11. distribute-lft-neg-out 74.8%

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

   12. remove-double-neg 74.8%

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

   13. associate-*r* 74.9%

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

   14. *-commutative 74.9%

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

   15. associate-*r* 72.0%

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

3. Simplified 76.8%

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

4. Taylor expanded in x around 0 51.1%

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

   1. unpow2 51.1%

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

6. Simplified 51.1%

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

7. Taylor expanded in x around inf 31.8%

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

   1. unpow2 31.8%

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

   2. unpow2 31.8%

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

   3. associate-/r* 31.7%

      \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot c}}{s \cdot s}} \]

9. Simplified 31.7%

   \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot c}}{s \cdot s}} \]

10. Taylor expanded in c around 0 31.8%

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

    1. unpow2 31.8%

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

    2. unpow2 31.8%

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

    3. *-commutative 31.8%

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

12. Simplified 31.8%

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

13. Final simplification 31.8%

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

Reproduce

herbie shell --seed 2023293 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))