mixedcos

Percentage Accurate: 67.2% → 96.9%

Time: 22.6s

Alternatives: 20

Speedup: 24.1×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(s \cdot x\right)\\ \frac{1}{t_0} \cdot \frac{{\cos x}^{4} - {\sin x}^{4}}{t_0} \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s x))))
   (* (/ 1.0 t_0) (/ (- (pow (cos x) 4.0) (pow (sin x) 4.0)) t_0))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (s * x);
	return (1.0 / t_0) * ((pow(cos(x), 4.0) - pow(sin(x), 4.0)) / t_0);
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (s * x)
    code = (1.0d0 / t_0) * (((cos(x) ** 4.0d0) - (sin(x) ** 4.0d0)) / t_0)
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (s * x);
	return (1.0 / t_0) * ((Math.pow(Math.cos(x), 4.0) - Math.pow(Math.sin(x), 4.0)) / t_0);
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (s * x)
	return (1.0 / t_0) * ((math.pow(math.cos(x), 4.0) - math.pow(math.sin(x), 4.0)) / t_0)

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(s * x))
	return Float64(Float64(1.0 / t_0) * Float64(Float64((cos(x) ^ 4.0) - (sin(x) ^ 4.0)) / t_0))
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (s * x);
	tmp = (1.0 / t_0) * (((cos(x) ^ 4.0) - (sin(x) ^ 4.0)) / t_0);
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision] - N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. cos-2 98.0%

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

   3. flip-- 98.0%

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

   4. pow2 98.0%

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

   5. pow2 98.0%

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

   6. pow2 98.0%

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

   7. pow2 98.0%

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

   8. pow2 98.0%

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

   9. pow2 98.0%

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

7. Applied egg-rr 98.0%

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

   1. pow-sqr 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{\color{blue}{4}} - {\sin x}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2} + {\sin x}^{2}}}{c \cdot \left(s \cdot x\right)} \]

   3. pow-sqr 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{4} - \color{blue}{{\sin x}^{\left(2 \cdot 2\right)}}}{{\cos x}^{2} + {\sin x}^{2}}}{c \cdot \left(s \cdot x\right)} \]

   4. metadata-eval 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{4} - {\sin x}^{\color{blue}{4}}}{{\cos x}^{2} + {\sin x}^{2}}}{c \cdot \left(s \cdot x\right)} \]

   5. unpow2 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{4} - {\sin x}^{4}}{\color{blue}{\cos x \cdot \cos x} + {\sin x}^{2}}}{c \cdot \left(s \cdot x\right)} \]

   6. unpow2 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{4} - {\sin x}^{4}}{\cos x \cdot \cos x + \color{blue}{\sin x \cdot \sin x}}}{c \cdot \left(s \cdot x\right)} \]

   7. cos-sin-sum 98.0%

      \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{{\cos x}^{4} - {\sin x}^{4}}{\color{blue}{1}}}{c \cdot \left(s \cdot x\right)} \]

9. Simplified 98.0%

   \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\color{blue}{\frac{{\cos x}^{4} - {\sin x}^{4}}{1}}}{c \cdot \left(s \cdot x\right)} \]

10. Final simplification 98.0%

    \[\leadsto \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{{\cos x}^{4} - {\sin x}^{4}}{c \cdot \left(s \cdot x\right)} \]

Alternative 2: 84.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 6 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{t_0}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 6e-40)
     (* (/ (/ 1.0 c) (* s x)) (* (/ 1.0 c) (/ (/ 1.0 s) x)))
     (if (<= x 5e+30)
       (/ t_0 (* (* x x) (* (* c s) (* c s))))
       (/ t_0 (* x (* (* s s) (* c (* c x)))))))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 6e-40) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else if (x <= 5e+30) {
		tmp = t_0 / ((x * x) * ((c * s) * (c * s)));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (c * x))));
	}
	return tmp;
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 6d-40) then
        tmp = ((1.0d0 / c) / (s * x)) * ((1.0d0 / c) * ((1.0d0 / s) / x))
    else if (x <= 5d+30) then
        tmp = t_0 / ((x * x) * ((c * s) * (c * s)))
    else
        tmp = t_0 / (x * ((s * s) * (c * (c * x))))
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 6e-40) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else if (x <= 5e+30) {
		tmp = t_0 / ((x * x) * ((c * s) * (c * s)));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (c * x))));
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 6e-40:
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x))
	elif x <= 5e+30:
		tmp = t_0 / ((x * x) * ((c * s) * (c * s)))
	else:
		tmp = t_0 / (x * ((s * s) * (c * (c * x))))
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 6e-40)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(s * x)) * Float64(Float64(1.0 / c) * Float64(Float64(1.0 / s) / x)));
	elseif (x <= 5e+30)
		tmp = Float64(t_0 / Float64(Float64(x * x) * Float64(Float64(c * s) * Float64(c * s))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(s * s) * Float64(c * Float64(c * x)))));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 6e-40)
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	elseif (x <= 5e+30)
		tmp = t_0 / ((x * x) * ((c * s) * (c * s)));
	else
		tmp = t_0 / (x * ((s * s) * (c * (c * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6e-40], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+30], N[(t$95$0 / N[(N[(x * x), $MachinePrecision] * N[(N[(c * s), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.00000000000000039e-40

   1. Initial program 63.5%

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

      1. associate-/r* 63.5%

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

      2. remove-double-neg 63.5%

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

      3. distribute-lft-neg-out 63.5%

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

      4. distribute-lft-neg-out 63.5%

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

      5. distribute-rgt-neg-out 63.5%

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

      6. associate-/r* 63.5%

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

      7. unpow2 63.5%

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

      8. *-commutative 63.5%

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

      9. associate-*r* 55.9%

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

      10. sqr-neg 55.9%

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

      11. associate-*r* 63.5%

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

      12. unpow2 63.5%

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

   3. Simplified 63.5%

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

      1. associate-*r* 55.9%

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

      2. *-commutative 55.9%

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

      3. associate-*l* 57.0%

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

      4. swap-sqr 74.4%

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

      5. *-commutative 74.4%

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

      6. add-sqr-sqrt 74.4%

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

      7. *-un-lft-identity 74.4%

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

      8. times-frac 74.4%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in x around 0 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-/r* 85.5%

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

   8. Simplified 85.5%

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

      1. *-un-lft-identity 85.5%

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

      2. associate-/r* 85.5%

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

   10. Applied egg-rr 85.5%

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

       1. div-inv 85.5%

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

       2. associate-/r* 85.5%

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

   12. Applied egg-rr 85.5%

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

   if 6.00000000000000039e-40 < x < 4.9999999999999998e30

   1. Initial program 90.8%

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

      1. associate-/r* 90.8%

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

      2. *-commutative 90.8%

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

      3. associate-*l* 90.8%

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

      4. unpow2 90.8%

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

      5. unpow2 90.8%

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

      6. associate-*r* 90.8%

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

      7. associate-/r* 91.1%

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

      8. associate-/l/ 91.1%

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

      9. associate-/l/ 91.2%

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

      10. *-commutative 91.2%

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

      11. associate-*l* 91.4%

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

      12. unpow2 91.4%

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

      13. associate-*l* 90.8%

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

      14. unpow2 90.8%

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

      15. unswap-sqr 99.9%

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

      16. *-commutative 99.9%

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

      17. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 4.9999999999999998e30 < x

   1. Initial program 61.3%

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

      1. associate-/r* 59.7%

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

      2. remove-double-neg 59.7%

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

      3. distribute-lft-neg-out 59.7%

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

      4. distribute-lft-neg-out 59.7%

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

      5. distribute-rgt-neg-out 59.7%

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

      6. associate-/l/ 61.3%

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

      7. distribute-rgt-neg-out 61.3%

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

      8. distribute-lft-neg-out 61.3%

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

      9. associate-*l* 62.7%

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

      10. distribute-lft-neg-in 62.7%

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

      11. distribute-lft-neg-out 62.7%

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

      12. remove-double-neg 62.7%

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

      13. associate-*r* 62.8%

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

      14. *-commutative 62.8%

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

      15. associate-*r* 61.2%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in x around 0 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. associate-*r* 72.5%

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

   6. Simplified 72.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 3: 82.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 5e-43)
   (* (/ (/ 1.0 c) (* s x)) (* (/ 1.0 c) (/ (/ 1.0 s) x)))
   (/ (cos (* x 2.0)) (* x (* x (* c (* c (* s s))))))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-43) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Fortran

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 <= 5d-43) then
        tmp = ((1.0d0 / c) / (s * x)) * ((1.0d0 / c) * ((1.0d0 / s) / x))
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (c * (s * s)))))
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-43) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 5e-43:
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x))
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))))
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 5e-43)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(s * x)) * Float64(Float64(1.0 / c) * Float64(Float64(1.0 / s) / x)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5e-43)
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

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, 5e-43], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000019e-43

   1. Initial program 63.3%

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

      1. associate-/r* 63.3%

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

      2. remove-double-neg 63.3%

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

      3. distribute-lft-neg-out 63.3%

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

      4. distribute-lft-neg-out 63.3%

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

      5. distribute-rgt-neg-out 63.3%

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

      6. associate-/r* 63.3%

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

      7. unpow2 63.3%

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

      8. *-commutative 63.3%

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

      9. associate-*r* 55.6%

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

      10. sqr-neg 55.6%

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

      11. associate-*r* 63.3%

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

      12. unpow2 63.3%

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

   3. Simplified 63.3%

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

      1. associate-*r* 55.6%

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

      2. *-commutative 55.6%

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

      3. associate-*l* 56.8%

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

      4. swap-sqr 74.3%

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

      5. *-commutative 74.3%

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

      6. add-sqr-sqrt 74.3%

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

      7. *-un-lft-identity 74.3%

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

      8. times-frac 74.3%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in x around 0 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-/r* 85.4%

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

   8. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. associate-/r* 85.4%

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

   10. Applied egg-rr 85.4%

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

       1. div-inv 85.4%

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

       2. associate-/r* 85.4%

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

   12. Applied egg-rr 85.4%

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

   if 5.00000000000000019e-43 < x

   1. Initial program 66.3%

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

      1. associate-/r* 64.9%

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

      2. remove-double-neg 64.9%

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

      3. distribute-lft-neg-out 64.9%

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

      4. distribute-lft-neg-out 64.9%

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

      5. distribute-rgt-neg-out 64.9%

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

      6. associate-/l/ 66.3%

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

      7. distribute-rgt-neg-out 66.3%

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

      8. distribute-lft-neg-out 66.3%

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

      9. associate-*l* 67.5%

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

      10. distribute-lft-neg-in 67.5%

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

      11. distribute-lft-neg-out 67.5%

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

      12. remove-double-neg 67.5%

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

      13. associate-*r* 67.5%

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

      14. *-commutative 67.5%

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

      15. associate-*r* 66.1%

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

   3. Simplified 74.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4: 84.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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 5e-43)
   (* (/ (/ 1.0 c) (* s x)) (* (/ 1.0 c) (/ (/ 1.0 s) x)))
   (/ (cos (* x 2.0)) (* x (* x (* c (* s (* c s))))))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-43) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))));
	}
	return tmp;
}

Fortran

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 <= 5d-43) then
        tmp = ((1.0d0 / c) / (s * x)) * ((1.0d0 / c) * ((1.0d0 / s) / x))
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (s * (c * s)))))
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-43) {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))));
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 5e-43:
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x))
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))))
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 5e-43)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(s * x)) * Float64(Float64(1.0 / c) * Float64(Float64(1.0 / s) / x)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5e-43)
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

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, 5e-43], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000019e-43

   1. Initial program 63.3%

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

      1. associate-/r* 63.3%

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

      2. remove-double-neg 63.3%

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

      3. distribute-lft-neg-out 63.3%

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

      4. distribute-lft-neg-out 63.3%

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

      5. distribute-rgt-neg-out 63.3%

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

      6. associate-/r* 63.3%

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

      7. unpow2 63.3%

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

      8. *-commutative 63.3%

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

      9. associate-*r* 55.6%

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

      10. sqr-neg 55.6%

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

      11. associate-*r* 63.3%

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

      12. unpow2 63.3%

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

   3. Simplified 63.3%

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

      1. associate-*r* 55.6%

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

      2. *-commutative 55.6%

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

      3. associate-*l* 56.8%

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

      4. swap-sqr 74.3%

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

      5. *-commutative 74.3%

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

      6. add-sqr-sqrt 74.3%

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

      7. *-un-lft-identity 74.3%

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

      8. times-frac 74.3%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in x around 0 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-/r* 85.4%

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

   8. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. associate-/r* 85.4%

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

   10. Applied egg-rr 85.4%

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

       1. div-inv 85.4%

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

       2. associate-/r* 85.4%

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

   12. Applied egg-rr 85.4%

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

   if 5.00000000000000019e-43 < x

   1. Initial program 66.3%

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

      1. associate-/r* 64.9%

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

      2. remove-double-neg 64.9%

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

      3. distribute-lft-neg-out 64.9%

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

      4. distribute-lft-neg-out 64.9%

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

      5. distribute-rgt-neg-out 64.9%

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

      6. associate-/l/ 66.3%

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

      7. distribute-rgt-neg-out 66.3%

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

      8. distribute-lft-neg-out 66.3%

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

      9. associate-*l* 67.5%

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

      10. distribute-lft-neg-in 67.5%

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

      11. distribute-lft-neg-out 67.5%

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

      12. remove-double-neg 67.5%

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

      13. associate-*r* 67.5%

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

      14. *-commutative 67.5%

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

      15. associate-*r* 66.1%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in c around 0 74.3%

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

      1. unpow2 56.8%

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

      2. associate-*r* 64.9%

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

   6. Simplified 86.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 87.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \end{array} \end{array} \]

FPCore

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 (<= s 9.5e+146)
   (/ (cos (* x 2.0)) (* x (* (* s s) (* c (* c x)))))
   (* (/ (/ 1.0 c) (* s x)) (* (/ 1.0 c) (/ (/ 1.0 s) x)))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (s <= 9.5e+146) {
		tmp = cos((x * 2.0)) / (x * ((s * s) * (c * (c * x))));
	} else {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	}
	return tmp;
}

Fortran

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 (s <= 9.5d+146) then
        tmp = cos((x * 2.0d0)) / (x * ((s * s) * (c * (c * x))))
    else
        tmp = ((1.0d0 / c) / (s * x)) * ((1.0d0 / c) * ((1.0d0 / s) / x))
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 9.5e+146) {
		tmp = Math.cos((x * 2.0)) / (x * ((s * s) * (c * (c * x))));
	} else {
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if s <= 9.5e+146:
		tmp = math.cos((x * 2.0)) / (x * ((s * s) * (c * (c * x))))
	else:
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x))
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (s <= 9.5e+146)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(s * s) * Float64(c * Float64(c * x)))));
	else
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(s * x)) * Float64(Float64(1.0 / c) * Float64(Float64(1.0 / s) / x)));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 9.5e+146)
		tmp = cos((x * 2.0)) / (x * ((s * s) * (c * (c * x))));
	else
		tmp = ((1.0 / c) / (s * x)) * ((1.0 / c) * ((1.0 / s) / x));
	end
	tmp_2 = tmp;
end

Wolfram

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[s, 9.5e+146], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 9.49999999999999926e146

   1. Initial program 65.0%

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

      1. associate-/r* 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-lft-neg-out 64.6%

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

      4. distribute-lft-neg-out 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. associate-/l/ 65.0%

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

      7. distribute-rgt-neg-out 65.0%

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

      8. distribute-lft-neg-out 65.0%

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

      9. associate-*l* 66.4%

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

      10. distribute-lft-neg-in 66.4%

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

      11. distribute-lft-neg-out 66.4%

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

      12. remove-double-neg 66.4%

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

      13. associate-*r* 67.3%

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

      14. *-commutative 67.3%

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

      15. associate-*r* 66.1%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around 0 67.3%

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

      1. unpow2 67.3%

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

      2. unpow2 67.3%

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

      3. associate-*r* 74.8%

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

   6. Simplified 74.8%

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

   if 9.49999999999999926e146 < s

   1. Initial program 55.0%

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

      1. associate-/r* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. distribute-lft-neg-out 54.6%

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

      4. distribute-lft-neg-out 54.6%

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

      5. distribute-rgt-neg-out 54.6%

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

      6. associate-/r* 55.0%

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

      7. unpow2 55.0%

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

      8. *-commutative 55.0%

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

      9. associate-*r* 45.8%

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

      10. sqr-neg 45.8%

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

      11. associate-*r* 55.0%

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

      12. unpow2 55.0%

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

   3. Simplified 55.0%

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

      1. associate-*r* 45.8%

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

      2. *-commutative 45.8%

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

      3. associate-*l* 45.8%

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

      4. swap-sqr 86.5%

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

      5. *-commutative 86.5%

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

      6. add-sqr-sqrt 86.5%

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

      7. *-un-lft-identity 86.5%

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

      8. times-frac 86.5%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-/r* 91.1%

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

   8. Simplified 91.1%

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

      1. *-un-lft-identity 91.1%

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

      2. associate-/r* 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. div-inv 91.1%

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

       2. associate-/r* 91.1%

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

   12. Applied egg-rr 91.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \left(\frac{1}{c} \cdot \frac{\frac{1}{s}}{x}\right)\\ \end{array} \]

Alternative 6: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(s \cdot x\right)\\ \frac{1}{t_0} \cdot \frac{\cos \left(x \cdot 2\right)}{t_0} \end{array} \end{array} \]

FPCore

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

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (s * x);
	return (1.0 / t_0) * (cos((x * 2.0)) / t_0);
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (s * x)
    code = (1.0d0 / t_0) * (cos((x * 2.0d0)) / t_0)
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (s * x);
	return (1.0 / t_0) * (Math.cos((x * 2.0)) / t_0);
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (s * x)
	return (1.0 / t_0) * (math.cos((x * 2.0)) / t_0)

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(s * x))
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x * 2.0)) / t_0))
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (s * x);
	tmp = (1.0 / t_0) * (cos((x * 2.0)) / t_0);
end

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot x}}{c} \end{array} \]

FPCore

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))) (/ (/ (cos (* x 2.0)) (* s x)) c)))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return (1.0 / (c * (s * x))) * ((cos((x * 2.0)) / (s * x)) / c);
}

Fortran

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))) * ((cos((x * 2.0d0)) / (s * x)) / c)
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return (1.0 / (c * (s * x))) * ((Math.cos((x * 2.0)) / (s * x)) / c);
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return (1.0 / (c * (s * x))) * ((math.cos((x * 2.0)) / (s * x)) / c)

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(Float64(1.0 / Float64(c * Float64(s * x))) * Float64(Float64(cos(Float64(x * 2.0)) / Float64(s * x)) / c))
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = (1.0 / (c * (s * x))) * ((cos((x * 2.0)) / (s * x)) / c);
end

Wolfram

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[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

   1. *-un-lft-identity 97.7%

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

   2. times-frac 97.7%

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

7. Applied egg-rr 97.7%

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

   1. associate-*l/ 97.8%

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

   2. *-un-lft-identity 97.8%

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

   3. *-commutative 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

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

Alternative 8: 79.1% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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)) * ((1.0d0 / c) * ((1.0d0 / s) / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 80.0%

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

   1. *-commutative 80.0%

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

   2. associate-/r* 80.0%

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

8. Simplified 80.0%

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

   1. *-un-lft-identity 80.0%

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

   2. associate-/r* 80.0%

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

10. Applied egg-rr 80.0%

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

    1. div-inv 80.0%

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

    2. associate-/r* 80.0%

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

12. Applied egg-rr 80.0%

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

13. Final simplification 80.0%

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

Alternative 9: 73.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;c \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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 (<= c 6e-77)
   (/ 1.0 (* x (* x (* c (* s (* c s))))))
   (/ 1.0 (* s (* x (* s (* c (* c x))))))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (c <= 6e-77) {
		tmp = 1.0 / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = 1.0 / (s * (x * (s * (c * (c * x)))));
	}
	return tmp;
}

Fortran

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 (c <= 6d-77) then
        tmp = 1.0d0 / (x * (x * (c * (s * (c * s)))))
    else
        tmp = 1.0d0 / (s * (x * (s * (c * (c * x)))))
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 6e-77) {
		tmp = 1.0 / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = 1.0 / (s * (x * (s * (c * (c * x)))));
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if c <= 6e-77:
		tmp = 1.0 / (x * (x * (c * (s * (c * s)))))
	else:
		tmp = 1.0 / (s * (x * (s * (c * (c * x)))))
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (c <= 6e-77)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	else
		tmp = Float64(1.0 / Float64(s * Float64(x * Float64(s * Float64(c * Float64(c * x))))));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 6e-77)
		tmp = 1.0 / (x * (x * (c * (s * (c * s)))));
	else
		tmp = 1.0 / (s * (x * (s * (c * (c * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

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[c, 6e-77], N[(1.0 / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(x * N[(s * N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 6.00000000000000033e-77

   1. Initial program 66.4%

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

      1. associate-/r* 65.8%

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

      2. remove-double-neg 65.8%

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

      3. distribute-lft-neg-out 65.8%

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

      4. distribute-lft-neg-out 65.8%

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

      5. distribute-rgt-neg-out 65.8%

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

      6. associate-/l/ 66.4%

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

      7. distribute-rgt-neg-out 66.4%

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

      8. distribute-lft-neg-out 66.4%

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

      9. associate-*l* 67.5%

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

      10. distribute-lft-neg-in 67.5%

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

      11. distribute-lft-neg-out 67.5%

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

      12. remove-double-neg 67.5%

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

      13. associate-*r* 68.2%

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

      14. *-commutative 68.2%

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

      15. associate-*r* 67.7%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in x around 0 67.6%

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

   5. Taylor expanded in c around 0 67.6%

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

      1. unpow2 67.6%

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

      2. associate-*r* 74.8%

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

   7. Simplified 74.8%

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

   if 6.00000000000000033e-77 < c

   1. Initial program 59.1%

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

      1. associate-/r* 59.3%

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

      2. remove-double-neg 59.3%

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

      3. distribute-lft-neg-out 59.3%

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

      4. distribute-lft-neg-out 59.3%

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

      5. distribute-rgt-neg-out 59.3%

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

      6. associate-/l/ 59.1%

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

      7. distribute-rgt-neg-out 59.1%

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

      8. distribute-lft-neg-out 59.1%

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

      9. associate-*l* 60.7%

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

      10. distribute-lft-neg-in 60.7%

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

      11. distribute-lft-neg-out 60.7%

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

      12. remove-double-neg 60.7%

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

      13. associate-*r* 62.0%

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

      14. *-commutative 62.0%

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

      15. associate-*r* 59.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*r* 52.7%

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

      3. associate-*r* 55.0%

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

      4. *-commutative 55.0%

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

      5. associate-*r* 53.8%

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

      6. *-commutative 53.8%

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

      7. associate-*r* 52.2%

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

      8. add-sqr-sqrt 52.2%

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

      9. pow2 52.2%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*r* 83.5%

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

      2. *-commutative 83.5%

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

      3. unpow-prod-down 66.5%

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

      4. pow2 66.5%

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

      5. associate-*l* 71.4%

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

      6. unpow2 71.4%

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

      7. swap-sqr 52.7%

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

      8. associate-*r* 55.0%

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

      9. associate-*r* 63.6%

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

      10. associate-*l* 70.3%

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

      11. associate-*l* 71.7%

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

   8. Applied egg-rr 71.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10: 79.1% accurate, 20.9× speedup

Math

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

FPCore

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 (/ 1.0 (* c (* s x))))) (* t_0 t_0)))

C

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

Fortran

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 = 1.0d0 / (c * (s * x))
    code = t_0 * t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[(1.0 / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 80.0%

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

7. Final simplification 80.0%

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

Alternative 11: 79.2% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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))) * ((1.0d0 / c) / (s * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

   1. *-un-lft-identity 97.7%

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

   2. times-frac 97.7%

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

7. Applied egg-rr 97.7%

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

   1. associate-*l/ 97.8%

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

   2. *-un-lft-identity 97.8%

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

   3. *-commutative 97.8%

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

9. Applied egg-rr 97.8%

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

10. Taylor expanded in x around 0 80.0%

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

    1. associate-/r* 80.0%

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

12. Simplified 80.0%

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

13. Final simplification 80.0%

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

Alternative 12: 79.1% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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))) * (((1.0d0 / x) / s) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

   1. *-un-lft-identity 97.7%

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

   2. times-frac 97.7%

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

7. Applied egg-rr 97.7%

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

   1. associate-*l/ 97.8%

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

   2. *-un-lft-identity 97.8%

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

   3. *-commutative 97.8%

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

9. Applied egg-rr 97.8%

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

10. Taylor expanded in x around 0 80.0%

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

    1. associate-/l/ 80.0%

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

12. Simplified 80.0%

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

13. Final simplification 80.0%

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

Alternative 13: 79.1% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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))) * (((1.0d0 / s) / x) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/r* 64.1%

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

   7. unpow2 64.1%

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

   8. *-commutative 64.1%

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

   9. associate-*r* 58.2%

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

   10. sqr-neg 58.2%

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

   11. associate-*r* 64.1%

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

   12. unpow2 64.1%

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

3. Simplified 64.1%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*l* 59.0%

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

   4. swap-sqr 76.3%

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

   5. *-commutative 76.3%

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

   6. add-sqr-sqrt 76.3%

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

   7. *-un-lft-identity 76.3%

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

   8. times-frac 76.3%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 80.0%

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

   1. *-commutative 80.0%

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

   2. associate-/r* 80.0%

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

8. Simplified 80.0%

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

   1. *-un-lft-identity 80.0%

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

   2. associate-/r* 80.0%

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

10. Applied egg-rr 80.0%

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

11. Final simplification 80.0%

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

Alternative 14: 67.5% accurate, 24.1× speedup

Math

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

FPCore

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 (* (* c s) (* c (* x x))))))

C

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

Fortran

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 * ((c * s) * (c * (x * x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/l/ 64.1%

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

   7. distribute-rgt-neg-out 64.1%

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

   8. distribute-lft-neg-out 64.1%

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

   9. associate-*l* 65.4%

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

   10. distribute-lft-neg-in 65.4%

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

   11. distribute-lft-neg-out 65.4%

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

   12. remove-double-neg 65.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 65.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 65.7%

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

   1. *-commutative 65.7%

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

   2. associate-*r* 58.1%

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

   3. associate-*r* 58.9%

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

   4. *-commutative 58.9%

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

   5. associate-*r* 58.5%

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

   6. *-commutative 58.5%

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

   7. associate-*r* 57.5%

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

   8. add-sqr-sqrt 57.5%

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

   9. pow2 57.5%

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

6. Applied egg-rr 79.6%

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

   1. unpow2 79.6%

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

   2. associate-*r* 78.9%

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

   3. *-commutative 78.9%

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

   4. associate-*r* 80.2%

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

   5. *-commutative 80.2%

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

   6. swap-sqr 67.2%

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

   7. unpow2 67.2%

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

   8. associate-*l* 74.9%

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

   9. unpow2 74.9%

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

   10. swap-sqr 58.1%

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

   11. associate-*r* 58.9%

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

   12. associate-*r* 65.6%

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

   13. associate-*r* 68.1%

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

   14. associate-*l* 66.0%

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

   15. associate-*l* 72.7%

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

   16. *-commutative 72.7%

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

8. Applied egg-rr 72.7%

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

   1. associate-*l* 74.4%

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

   2. *-commutative 74.4%

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

   3. associate-*l* 67.8%

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

10. Simplified 67.8%

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

11. Final simplification 67.8%

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

Alternative 15: 72.3% accurate, 24.1× speedup

Math

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

FPCore

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 (* s (* c (* c x)))))))

C

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

Fortran

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 * (s * (c * (c * x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/l/ 64.1%

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

   7. distribute-rgt-neg-out 64.1%

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

   8. distribute-lft-neg-out 64.1%

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

   9. associate-*l* 65.4%

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

   10. distribute-lft-neg-in 65.4%

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

   11. distribute-lft-neg-out 65.4%

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

   12. remove-double-neg 65.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 65.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 65.7%

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

   1. *-commutative 65.7%

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

   2. associate-*r* 58.1%

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

   3. associate-*r* 58.9%

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

   4. *-commutative 58.9%

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

   5. associate-*r* 58.5%

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

   6. *-commutative 58.5%

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

   7. associate-*r* 57.5%

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

   8. add-sqr-sqrt 57.5%

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

   9. pow2 57.5%

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

6. Applied egg-rr 79.6%

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

   1. associate-*r* 80.2%

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

   2. *-commutative 80.2%

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

   3. unpow-prod-down 67.2%

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

   4. pow2 67.2%

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

   5. associate-*l* 74.9%

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

   6. unpow2 74.9%

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

   7. swap-sqr 58.1%

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

   8. associate-*r* 58.9%

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

   9. associate-*r* 65.6%

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

   10. associate-*l* 71.5%

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

   11. associate-*l* 72.1%

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

8. Applied egg-rr 72.1%

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

9. Final simplification 72.1%

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

Alternative 16: 76.4% accurate, 24.1× speedup

Math

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

FPCore

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 (* s x))))))

C

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

Fortran

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 * (s * x))))
end function

Java

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

Python

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

Julia

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(s * x)))))
end

MATLAB

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

Wolfram

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[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/l/ 64.1%

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

   7. distribute-rgt-neg-out 64.1%

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

   8. distribute-lft-neg-out 64.1%

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

   9. associate-*l* 65.4%

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

   10. distribute-lft-neg-in 65.4%

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

   11. distribute-lft-neg-out 65.4%

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

   12. remove-double-neg 65.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 65.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 65.7%

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

   1. *-commutative 65.7%

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

   2. associate-*r* 58.1%

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

   3. associate-*r* 58.9%

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

   4. *-commutative 58.9%

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

   5. associate-*r* 58.5%

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

   6. *-commutative 58.5%

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

   7. associate-*r* 57.5%

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

   8. add-sqr-sqrt 57.5%

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

   9. pow2 57.5%

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

6. Applied egg-rr 79.6%

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

   1. unpow2 79.6%

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

   2. associate-*r* 78.9%

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

   3. *-commutative 78.9%

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

   4. associate-*l* 77.4%

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

   5. *-commutative 77.4%

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

8. Applied egg-rr 77.4%

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

9. Final simplification 77.4%

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

Alternative 17: 79.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/l/ 64.1%

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

   7. distribute-rgt-neg-out 64.1%

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

   8. distribute-lft-neg-out 64.1%

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

   9. associate-*l* 65.4%

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

   10. distribute-lft-neg-in 65.4%

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

   11. distribute-lft-neg-out 65.4%

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

   12. remove-double-neg 65.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 65.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 65.7%

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

   1. *-commutative 65.7%

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

   2. associate-*r* 58.1%

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

   3. associate-*r* 58.9%

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

   4. *-commutative 58.9%

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

   5. associate-*r* 58.5%

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

   6. *-commutative 58.5%

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

   7. associate-*r* 57.5%

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

   8. add-sqr-sqrt 57.5%

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

   9. pow2 57.5%

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

6. Applied egg-rr 79.6%

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

   1. unpow2 79.6%

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

8. Applied egg-rr 79.6%

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

9. Final simplification 79.6%

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

Alternative 18: 77.8% accurate, 24.1× speedup

Math

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

FPCore

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 (* s x))))))

C

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

Fortran

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 * (s * x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. associate-/r* 63.8%

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

   2. remove-double-neg 63.8%

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

   3. distribute-lft-neg-out 63.8%

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

   4. distribute-lft-neg-out 63.8%

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

   5. distribute-rgt-neg-out 63.8%

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

   6. associate-/l/ 64.1%

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

   7. distribute-rgt-neg-out 64.1%

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

   8. distribute-lft-neg-out 64.1%

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

   9. associate-*l* 65.4%

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

   10. distribute-lft-neg-in 65.4%

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

   11. distribute-lft-neg-out 65.4%

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

   12. remove-double-neg 65.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 65.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 65.7%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 65.7%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right) \cdot x}} \]

   2. associate-*r* 58.1%

      \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(s \cdot s\right)\right)}\right) \cdot x} \]

   3. associate-*r* 58.9%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)} \cdot x} \]

   4. *-commutative 58.9%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(s \cdot s\right)\right) \cdot x} \]

   5. associate-*r* 58.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}} \]

   6. *-commutative 58.5%

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot s\right)\right)}} \]

   7. associate-*r* 57.5%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   8. add-sqr-sqrt 57.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)} \cdot \sqrt{\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}}} \]

   9. pow2 57.5%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}\right)}^{2}}} \]

6. Applied egg-rr 79.6%

   \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 79.6%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

   2. *-commutative 79.6%

      \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot c\right)}} \]

   3. associate-*r* 79.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)\right) \cdot c}} \]

8. Applied egg-rr 79.4%

   \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)\right) \cdot c}} \]

9. Final simplification 79.4%

   \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)} \]

Alternative 19: 23.8% accurate, 28.5× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{1}{\frac{\frac{c \cdot \left(-x\right)}{x}}{-c}} \end{array} \]

FPCore

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)) x) (- c))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 1.0 / (((c * -x) / x) / -c);
}

Fortran

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) / x) / -c)
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return 1.0 / (((c * -x) / x) / -c);
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0 / (((c * -x) / x) / -c)

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(c * Float64(-x)) / x) / Float64(-c)))
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 1.0 / (((c * -x) / x) / -c);
end

Wolfram

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[(N[(c * (-x)), $MachinePrecision] / x), $MachinePrecision] / (-c)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 63.8%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/l/ 64.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right) \cdot {c}^{2}}} \]

   7. distribute-rgt-neg-out 64.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(-\left(\left(-x\right) \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   8. distribute-lft-neg-out 64.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{-\left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   9. associate-*l* 65.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(x \cdot {c}^{2}\right)}} \]

   10. distribute-lft-neg-in 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(-\left(-x\right) \cdot {s}^{2}\right) \cdot \left(x \cdot {c}^{2}\right)}} \]

   11. distribute-lft-neg-out 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(-\color{blue}{\left(-x \cdot {s}^{2}\right)}\right) \cdot \left(x \cdot {c}^{2}\right)} \]

   12. remove-double-neg 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot \left(x \cdot {c}^{2}\right)} \]

   13. associate-*r* 66.3%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({s}^{2} \cdot \left(x \cdot {c}^{2}\right)\right)}} \]

   14. *-commutative 66.3%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left(x \cdot {c}^{2}\right) \cdot {s}^{2}\right)}} \]

   15. associate-*r* 65.2%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

3. Simplified 74.2%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}} \]

4. Taylor expanded in x around 0 65.7%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)} \]

5. Taylor expanded in c around 0 65.7%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left({s}^{2} \cdot c\right)}\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 65.7%

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot c\right)\right)\right)} \]

   2. associate-*r* 73.8%

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

7. Simplified 73.8%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

8. Taylor expanded in x around 0 58.9%

   \[\leadsto \frac{1}{x \cdot \color{blue}{\left({s}^{2} \cdot \left({c}^{2} \cdot x\right)\right)}} \]
9. Step-by-step derivation

   1. *-commutative 58.9%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot {s}^{2}\right)}} \]

   2. associate-*r* 58.5%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right)} \]

   4. unpow2 58.5%

      \[\leadsto \frac{1}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]

10. Simplified 58.5%

    \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

12. Applied egg-rr 22.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{-\frac{\left(c \cdot x\right) \cdot s}{x \cdot s}}{-c}}} \]
13. Step-by-step derivation

    1. times-frac 26.7%

       \[\leadsto \frac{1}{\frac{-\color{blue}{\frac{c \cdot x}{x} \cdot \frac{s}{s}}}{-c}} \]

    2. *-inverses 26.7%

       \[\leadsto \frac{1}{\frac{-\frac{c \cdot x}{x} \cdot \color{blue}{1}}{-c}} \]

    3. associate-*l/ 26.7%

       \[\leadsto \frac{1}{\frac{-\color{blue}{\frac{\left(c \cdot x\right) \cdot 1}{x}}}{-c}} \]

    4. *-rgt-identity 26.7%

       \[\leadsto \frac{1}{\frac{-\frac{\color{blue}{c \cdot x}}{x}}{-c}} \]

14. Simplified 26.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{-\frac{c \cdot x}{x}}{-c}}} \]

15. Final simplification 26.7%

    \[\leadsto \frac{1}{\frac{\frac{c \cdot \left(-x\right)}{x}}{-c}} \]

Alternative 20: 3.5% accurate, 313.0× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ 1 \end{array} \]

FPCore

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 = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 1.0;
}

Fortran

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
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return 1.0;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return 1.0
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 1.0;
end

Wolfram

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_] := 1.0

Derivation

1. Initial program 64.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 63.8%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/l/ 64.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right) \cdot {c}^{2}}} \]

   7. distribute-rgt-neg-out 64.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(-\left(\left(-x\right) \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   8. distribute-lft-neg-out 64.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{-\left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   9. associate-*l* 65.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(x \cdot {c}^{2}\right)}} \]

   10. distribute-lft-neg-in 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(-\left(-x\right) \cdot {s}^{2}\right) \cdot \left(x \cdot {c}^{2}\right)}} \]

   11. distribute-lft-neg-out 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(-\color{blue}{\left(-x \cdot {s}^{2}\right)}\right) \cdot \left(x \cdot {c}^{2}\right)} \]

   12. remove-double-neg 65.4%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot \left(x \cdot {c}^{2}\right)} \]

   13. associate-*r* 66.3%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({s}^{2} \cdot \left(x \cdot {c}^{2}\right)\right)}} \]

   14. *-commutative 66.3%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left(x \cdot {c}^{2}\right) \cdot {s}^{2}\right)}} \]

   15. associate-*r* 65.2%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

3. Simplified 74.2%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}} \]

4. Taylor expanded in x around 0 65.7%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)} \]

5. Taylor expanded in c around 0 65.7%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left({s}^{2} \cdot c\right)}\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 65.7%

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot c\right)\right)\right)} \]

   2. associate-*r* 73.8%

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

7. Simplified 73.8%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

8. Taylor expanded in x around 0 58.9%

   \[\leadsto \frac{1}{x \cdot \color{blue}{\left({s}^{2} \cdot \left({c}^{2} \cdot x\right)\right)}} \]
9. Step-by-step derivation

   1. *-commutative 58.9%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot {s}^{2}\right)}} \]

   2. associate-*r* 58.5%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right)} \]

   4. unpow2 58.5%

      \[\leadsto \frac{1}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]

10. Simplified 58.5%

    \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

12. Applied egg-rr 22.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{\frac{\left(c \cdot x\right) \cdot s}{x \cdot s}}}}} \]
13. Step-by-step derivation

    1. associate-/r/ 24.0%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{c}{\left(c \cdot x\right) \cdot s} \cdot \left(x \cdot s\right)}}} \]

    2. associate-*l/ 8.1%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{c \cdot \left(x \cdot s\right)}{\left(c \cdot x\right) \cdot s}}}} \]

    3. associate-*r* 2.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}}} \]

    4. *-inverses 3.6%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{1}}} \]

    5. metadata-eval 3.6%

       \[\leadsto \frac{1}{\color{blue}{1}} \]

14. Simplified 3.6%

    \[\leadsto \frac{1}{\color{blue}{1}} \]

15. Final simplification 3.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))