mixedcos

Percentage Accurate: 66.2% → 99.1%

Time: 13.7s

Alternatives: 7

Speedup: 24.1×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{x \cdot c}}{s}\\ \end{array} \end{array} \]

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 2.5e-7)
   (/ (/ (/ (/ 1.0 s) x) c) (* c (* x s)))
   (/ (/ (/ (cos (* x 2.0)) (* s (* x c))) (* x c)) 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 <= 2.5e-7) {
		tmp = (((1.0 / s) / x) / c) / (c * (x * s));
	} else {
		tmp = ((cos((x * 2.0)) / (s * (x * c))) / (x * c)) / 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 <= 2.5d-7) then
        tmp = (((1.0d0 / s) / x) / c) / (c * (x * s))
    else
        tmp = ((cos((x * 2.0d0)) / (s * (x * c))) / (x * c)) / 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 <= 2.5e-7) {
		tmp = (((1.0 / s) / x) / c) / (c * (x * s));
	} else {
		tmp = ((Math.cos((x * 2.0)) / (s * (x * c))) / (x * c)) / 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 <= 2.5e-7:
		tmp = (((1.0 / s) / x) / c) / (c * (x * s))
	else:
		tmp = ((math.cos((x * 2.0)) / (s * (x * c))) / (x * c)) / 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 <= 2.5e-7)
		tmp = Float64(Float64(Float64(Float64(1.0 / s) / x) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(x * c))) / Float64(x * c)) / 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 <= 2.5e-7)
		tmp = (((1.0 / s) / x) / c) / (c * (x * s));
	else
		tmp = ((cos((x * 2.0)) / (s * (x * c))) / (x * c)) / 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, 2.5e-7], N[(N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.49999999999999989e-7

   1. Initial program 68.1%

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

   3. Applied egg-rr 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. *-commutative 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. times-frac 97.8%

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

      4. *-commutative 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. associate-*l/ 97.8%

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

      2. *-lft-identity 97.8%

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

      3. *-commutative 97.8%

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

   9. Simplified 97.8%

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

   10. Taylor expanded in x around 0 84.3%

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

       1. associate-/r* 84.4%

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

   12. Simplified 84.4%

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

   if 2.49999999999999989e-7 < x

   1. Initial program 72.5%

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

   3. Applied egg-rr 96.0%

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

      1. inv-pow 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r* 94.5%

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

      4. unpow-prod-down 94.5%

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

      5. inv-pow 94.5%

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

   5. Applied egg-rr 94.5%

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

      1. *-commutative 94.5%

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

      2. unpow-1 94.5%

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

      3. associate-*r/ 94.5%

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

      4. *-rgt-identity 94.5%

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

   7. Simplified 94.5%

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

      1. associate-*r/ 94.5%

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

      2. associate-*r* 98.2%

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

      3. associate-/r* 98.3%

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

      4. clear-num 98.3%

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

      5. associate-*l/ 98.3%

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

      6. *-un-lft-identity 98.3%

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

      7. *-commutative 98.3%

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

      8. div-inv 98.3%

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

      9. *-commutative 98.3%

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

      10. clear-num 98.3%

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

      11. /-rgt-identity 98.3%

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

      12. associate-*l* 99.7%

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

      13. *-commutative 99.7%

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

      14. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{x \cdot c}}{s}\\ \end{array} \]

Alternative 2: 96.8% accurate, 2.7× speedup

Math

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

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 (* x 2.0)) 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 (cos((x * 2.0)) / 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 = (cos((x * 2.0d0)) / 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 (Math.cos((x * 2.0)) / 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 (math.cos((x * 2.0)) / 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(Float64(cos(Float64(x * 2.0)) / 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((x * 2.0)) / 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[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 69.2%

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

3. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 3: 79.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;c \leq 4 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}}{c}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x}}{c}}{t_0}\\ \end{array} \end{array} \]

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 (<= c 4e-254)
     (/ (/ (+ (* -2.0 (/ x s)) (/ 1.0 (* x s))) c) t_0)
     (/ (/ (/ (/ 1.0 s) x) c) 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 (c <= 4e-254) {
		tmp = (((-2.0 * (x / s)) + (1.0 / (x * s))) / c) / t_0;
	} else {
		tmp = (((1.0 / s) / x) / c) / 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 (c <= 4d-254) then
        tmp = ((((-2.0d0) * (x / s)) + (1.0d0 / (x * s))) / c) / t_0
    else
        tmp = (((1.0d0 / s) / x) / c) / 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 (c <= 4e-254) {
		tmp = (((-2.0 * (x / s)) + (1.0 / (x * s))) / c) / t_0;
	} else {
		tmp = (((1.0 / s) / x) / c) / 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 c <= 4e-254:
		tmp = (((-2.0 * (x / s)) + (1.0 / (x * s))) / c) / t_0
	else:
		tmp = (((1.0 / s) / x) / c) / 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 (c <= 4e-254)
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64(x / s)) + Float64(1.0 / Float64(x * s))) / c) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / s) / x) / c) / 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 (c <= 4e-254)
		tmp = (((-2.0 * (x / s)) + (1.0 / (x * s))) / c) / t_0;
	else
		tmp = (((1.0 / s) / x) / c) / 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[c, 4e-254], N[(N[(N[(N[(-2.0 * N[(x / s), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 3.9999999999999996e-254

   1. Initial program 70.2%

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

   3. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. *-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. *-commutative 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. times-frac 98.1%

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

      4. *-commutative 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.1%

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

      2. *-lft-identity 98.1%

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

      3. *-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in x around 0 71.7%

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

   if 3.9999999999999996e-254 < c

   1. Initial program 67.9%

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

   3. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. *-commutative 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. *-commutative 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. times-frac 96.5%

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

      4. *-commutative 96.5%

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

   7. Applied egg-rr 96.5%

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

      1. associate-*l/ 96.6%

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

      2. *-lft-identity 96.6%

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

      3. *-commutative 96.6%

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

   9. Simplified 96.6%

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

   10. Taylor expanded in x around 0 79.6%

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

       1. associate-/r* 79.7%

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

   12. Simplified 79.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x}}{c}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 4: 79.5% 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 69.2%

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

2. Taylor expanded in x around 0 58.1%

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

   1. unpow2 58.1%

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

   2. *-commutative 58.1%

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

   3. unpow2 58.1%

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

   4. unpow2 58.1%

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

   5. swap-sqr 67.8%

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

   6. swap-sqr 79.1%

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

   7. unpow2 79.1%

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

   8. *-commutative 79.1%

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

4. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. pow2 79.1%

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

6. Applied egg-rr 79.1%

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

7. Final simplification 79.1%

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

Alternative 5: 79.6% accurate, 24.1× speedup

Math

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

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 (/ (/ (/ 1.0 c) (* x s)) (* 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) / (x * s)) / (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) / (x * s)) / (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) / (x * s)) / (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) / (x * s)) / (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(Float64(Float64(1.0 / c) / Float64(x * s)) / 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) / (x * s)) / (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[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

3. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. times-frac 97.4%

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

   4. *-commutative 97.4%

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

7. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Taylor expanded in x around 0 79.3%

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

    1. associate-/r* 79.3%

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

12. Simplified 79.3%

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

13. Final simplification 79.3%

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

Alternative 6: 79.6% accurate, 24.1× speedup

Math

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

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 (/ (/ (/ 1.0 (* x s)) c) (* 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 / (x * s)) / c) / (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 / (x * s)) / c) / (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 / (x * s)) / c) / (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 / (x * s)) / c) / (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(Float64(Float64(1.0 / Float64(x * s)) / c) / 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 / (x * s)) / c) / (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[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

3. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. times-frac 97.4%

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

   4. *-commutative 97.4%

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

7. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Taylor expanded in x around 0 79.4%

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

11. Final simplification 79.4%

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

Alternative 7: 79.6% accurate, 24.1× speedup

Math

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

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 (/ (/ (/ (/ 1.0 s) x) c) (* 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 / s) / x) / c) / (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 / s) / x) / c) / (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 / s) / x) / c) / (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 / s) / x) / c) / (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(Float64(Float64(Float64(1.0 / s) / x) / c) / 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 / s) / x) / c) / (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[(N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

3. Applied egg-rr 97.3%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. times-frac 97.4%

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

   4. *-commutative 97.4%

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

7. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Taylor expanded in x around 0 79.4%

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

    1. associate-/r* 79.4%

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

12. Simplified 79.4%

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

13. Final simplification 79.4%

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))