mixedcos

Percentage Accurate: 66.7% → 99.6%

Time: 16.9s

Alternatives: 11

Speedup: 24.1×

• 31.2% 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: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-34}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{1}{s}}{x}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot c}{\frac{1}{s}}} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}\\ \end{array} \end{array} \]

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 1e-34)
   (* (/ 1.0 (* c (* x s))) (/ (/ (/ 1.0 s) x) c))
   (* (/ 1.0 (/ (* x c) (/ 1.0 s))) (/ (/ (cos (* x 2.0)) s) (* x c)))))

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 <= 1e-34) {
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	} else {
		tmp = (1.0 / ((x * c) / (1.0 / s))) * ((cos((x * 2.0)) / s) / (x * c));
	}
	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 <= 1d-34) then
        tmp = (1.0d0 / (c * (x * s))) * (((1.0d0 / s) / x) / c)
    else
        tmp = (1.0d0 / ((x * c) / (1.0d0 / s))) * ((cos((x * 2.0d0)) / s) / (x * c))
    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 <= 1e-34) {
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	} else {
		tmp = (1.0 / ((x * c) / (1.0 / s))) * ((Math.cos((x * 2.0)) / s) / (x * c));
	}
	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 <= 1e-34:
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c)
	else:
		tmp = (1.0 / ((x * c) / (1.0 / s))) * ((math.cos((x * 2.0)) / s) / (x * c))
	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 <= 1e-34)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(Float64(1.0 / s) / x) / c));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x * c) / Float64(1.0 / s))) * Float64(Float64(cos(Float64(x * 2.0)) / s) / Float64(x * c)));
	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 <= 1e-34)
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	else
		tmp = (1.0 / ((x * c) / (1.0 / s))) * ((cos((x * 2.0)) / s) / (x * c));
	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, 1e-34], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x * c), $MachinePrecision] / N[(1.0 / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999928e-35

   1. Initial program 63.7%

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

   3. Applied egg-rr 95.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in c around 0 77.6%

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

      1. associate-/r* 77.7%

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

      2. *-rgt-identity 77.7%

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

      3. associate-*r/ 77.4%

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

      4. associate-*l/ 77.5%

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

      5. *-lft-identity 77.5%

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

      6. associate-/r* 77.5%

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

   9. Simplified 77.5%

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

   if 9.99999999999999928e-35 < x

   1. Initial program 67.8%

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

   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. *-un-lft-identity 98.1%

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

      2. associate-*r* 98.1%

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

      3. times-frac 98.1%

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

      4. *-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.2%

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

      2. *-lft-identity 98.2%

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

   7. Simplified 98.2%

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

      1. /-rgt-identity 98.2%

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

      2. associate-*r* 99.7%

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

      3. associate-/l* 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-34}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{1}{s}}{x}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot c}{\frac{1}{s}}} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.5× speedup

Math

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

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
 (* (cos (* x 2.0)) (pow (* x (* c s)) -2.0)))

C

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

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
    code = cos((x * 2.0d0)) * ((x * (c * s)) ** (-2.0d0))
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 Math.cos((x * 2.0)) * Math.pow((x * (c * s)), -2.0);
}

Python

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

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = cos((x * 2.0)) * ((x * (c * s)) ^ -2.0);
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_] := N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.7%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. associate-*r* 58.8%

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

   4. unpow2 58.8%

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

   5. associate-/l/ 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in x around inf 59.6%

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

   1. unpow2 59.6%

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

   2. unpow2 59.6%

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

   3. swap-sqr 76.5%

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

   4. unpow2 76.5%

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

6. Simplified 76.5%

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

   1. div-inv 76.2%

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

   2. div-inv 76.2%

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

   3. associate-*l* 76.2%

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

   4. *-commutative 76.2%

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

   5. pow-flip 76.2%

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

   6. metadata-eval 76.2%

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

   7. pow-flip 76.4%

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

   8. metadata-eval 76.4%

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

   9. unpow-prod-down 98.2%

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

   10. associate-*r* 96.1%

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

   11. *-commutative 96.1%

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

   12. *-commutative 96.1%

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

   13. *-commutative 96.1%

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

8. Applied egg-rr 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-*r* 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 3: 98.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{1}{s}}{x}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 2e-33)
   (* (/ 1.0 (* c (* x s))) (/ (/ (/ 1.0 s) x) c))
   (/ (/ (cos (* x 2.0)) s) (* (* x c) (* s (* x c))))))

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 <= 2e-33) {
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	} else {
		tmp = (cos((x * 2.0)) / s) / ((x * c) * (s * (x * c)));
	}
	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 <= 2d-33) then
        tmp = (1.0d0 / (c * (x * s))) * (((1.0d0 / s) / x) / c)
    else
        tmp = (cos((x * 2.0d0)) / s) / ((x * c) * (s * (x * c)))
    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 <= 2e-33) {
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	} else {
		tmp = (Math.cos((x * 2.0)) / s) / ((x * c) * (s * (x * c)));
	}
	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 <= 2e-33:
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c)
	else:
		tmp = (math.cos((x * 2.0)) / s) / ((x * c) * (s * (x * c)))
	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 <= 2e-33)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(Float64(1.0 / s) / x) / c));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / s) / Float64(Float64(x * c) * Float64(s * Float64(x * c))));
	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 <= 2e-33)
		tmp = (1.0 / (c * (x * s))) * (((1.0 / s) / x) / c);
	else
		tmp = (cos((x * 2.0)) / s) / ((x * c) * (s * (x * c)));
	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, 2e-33], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-33

   1. Initial program 63.7%

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

   3. Applied egg-rr 95.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in c around 0 77.6%

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

      1. associate-/r* 77.7%

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

      2. *-rgt-identity 77.7%

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

      3. associate-*r/ 77.4%

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

      4. associate-*l/ 77.5%

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

      5. *-lft-identity 77.5%

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

      6. associate-/r* 77.5%

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

   9. Simplified 77.5%

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

   if 2.0000000000000001e-33 < x

   1. Initial program 67.8%

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

   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. *-un-lft-identity 98.1%

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

      2. associate-*r* 98.1%

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

      3. times-frac 98.1%

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

      4. *-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.2%

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

      2. *-lft-identity 98.2%

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

   7. Simplified 98.2%

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

      1. /-rgt-identity 98.2%

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

      2. associate-*r* 99.7%

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

      3. associate-/l* 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. frac-times 96.6%

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

       2. *-un-lft-identity 96.6%

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

       3. div-inv 96.5%

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

       4. inv-pow 96.5%

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

       5. pow-flip 96.6%

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

       6. metadata-eval 96.6%

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

       7. pow1 96.6%

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

   11. Applied egg-rr 96.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{1}{s}}{x}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 4: 93.7% accurate, 2.7× speedup

Math

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

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

Derivation

1. Initial program 64.7%

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

3. Applied egg-rr 96.1%

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

   1. associate-/r* 96.1%

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

   2. frac-times 93.1%

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

   3. *-un-lft-identity 93.1%

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

   4. *-commutative 93.1%

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

5. Applied egg-rr 93.1%

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

6. Final simplification 93.1%

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

Alternative 5: 97.1% 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 64.7%

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

3. Applied egg-rr 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-*l/ 96.1%

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

   3. div-inv 96.2%

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

   4. *-commutative 96.2%

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

5. Applied egg-rr 96.2%

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

6. Final simplification 96.2%

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

Alternative 6: 79.8% accurate, 20.9× speedup

Math

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

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))) (/ (/ (/ 1.0 s) x) c)))

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))) * (((1.0 / s) / x) / c);
}

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))) * (((1.0d0 / s) / x) / c)
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))) * (((1.0 / s) / x) / c);
}

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))) * (((1.0 / s) / x) / c)

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(Float64(1.0 / s) / x) / c))
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))) * (((1.0 / s) / x) / c);
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[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.7%

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

3. Applied egg-rr 96.1%

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

4. Taylor expanded in x around 0 75.4%

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

   1. associate-*r* 74.5%

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

6. Simplified 74.5%

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

7. Taylor expanded in c around 0 75.4%

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

   1. associate-/r* 75.5%

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

   2. *-rgt-identity 75.5%

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

   3. associate-*r/ 75.3%

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

   4. associate-*l/ 75.3%

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

   5. *-lft-identity 75.3%

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

   6. associate-/r* 75.3%

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

9. Simplified 75.3%

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

10. Final simplification 75.3%

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

Alternative 7: 76.4% 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 64.7%

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

2. Taylor expanded in x around 0 52.4%

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

   1. associate-/r* 52.4%

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

   2. *-commutative 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 62.1%

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

   8. unpow2 62.1%

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

   9. unpow2 62.1%

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

   10. swap-sqr 75.4%

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

   11. unpow2 75.4%

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

   12. *-commutative 75.4%

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

4. Simplified 75.4%

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

   1. unpow2 75.4%

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

   2. associate-*r* 74.6%

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

   3. *-commutative 74.6%

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

   4. associate-*l* 73.2%

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

6. Applied egg-rr 73.2%

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

7. Final simplification 73.2%

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

Alternative 8: 76.5% accurate, 24.1× speedup

Math

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

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

Derivation

1. Initial program 64.7%

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

2. Taylor expanded in x around 0 52.4%

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

   1. associate-/r* 52.4%

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

   2. *-commutative 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 62.1%

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

   8. unpow2 62.1%

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

   9. unpow2 62.1%

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

   10. swap-sqr 75.4%

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

   11. unpow2 75.4%

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

   12. *-commutative 75.4%

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

4. Simplified 75.4%

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

   1. unpow2 75.4%

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

   2. associate-*r* 74.6%

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

   3. *-commutative 74.6%

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

   4. associate-*l* 73.2%

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

6. Applied egg-rr 73.2%

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

7. Taylor expanded in c around 0 73.2%

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

   1. associate-*r* 75.1%

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

9. Simplified 75.1%

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

10. Final simplification 75.1%

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

Alternative 9: 78.0% accurate, 24.1× speedup

Math

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

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

Derivation

1. Initial program 64.7%

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

2. Taylor expanded in x around 0 52.4%

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

   1. associate-/r* 52.4%

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

   2. *-commutative 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 62.1%

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

   8. unpow2 62.1%

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

   9. unpow2 62.1%

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

   10. swap-sqr 75.4%

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

   11. unpow2 75.4%

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

   12. *-commutative 75.4%

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

4. Simplified 75.4%

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

   1. unpow-prod-down 62.1%

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

   2. *-commutative 62.1%

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

   3. unpow-prod-down 75.4%

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

   4. unpow2 75.4%

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

   5. associate-*r* 73.4%

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

6. Applied egg-rr 73.4%

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

7. Final simplification 73.4%

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

Alternative 10: 79.0% accurate, 24.1× speedup

Math

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

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

Derivation

1. Initial program 64.7%

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

2. Taylor expanded in x around 0 52.4%

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

   1. associate-/r* 52.4%

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

   2. *-commutative 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 62.1%

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

   8. unpow2 62.1%

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

   9. unpow2 62.1%

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

   10. swap-sqr 75.4%

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

   11. unpow2 75.4%

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

   12. *-commutative 75.4%

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

4. Simplified 75.4%

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

   1. unpow-prod-down 62.1%

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

   2. *-commutative 62.1%

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

   3. unpow-prod-down 75.4%

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

   4. unpow2 75.4%

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

   5. *-commutative 75.4%

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

   6. associate-*r* 74.4%

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

6. Applied egg-rr 74.4%

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

7. Final simplification 74.4%

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

Alternative 11: 79.7% accurate, 24.1× speedup

Math

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

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)))) (/ (/ 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(Float64(1.0 / 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 = (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[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

3. Applied egg-rr 96.1%

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

4. Taylor expanded in x around 0 75.4%

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

   1. associate-*r* 74.5%

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

6. Simplified 74.5%

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

7. Taylor expanded in c around 0 74.5%

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

   1. associate-*r* 75.1%

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

9. Simplified 77.0%

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

    1. un-div-inv 77.0%

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

    2. associate-*l* 74.6%

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

    3. *-commutative 74.6%

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

    4. associate-*l* 75.5%

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

    5. *-commutative 75.5%

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

11. Applied egg-rr 75.5%

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

12. Final simplification 75.5%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))