mixedcos

Percentage Accurate: 66.3% → 99.0%

Time: 12.5s

Alternatives: 14

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: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\cos x}^{2} - {\sin x}^{2}}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 5e+47)
   (* (/ 1.0 (pow (* c (* x s)) 2.0)) (cos (* x 2.0)))
   (/ (- (pow (cos x) 2.0) (pow (sin x) 2.0)) (pow (* s (* x c)) 2.0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e+47) {
		tmp = (1.0 / pow((c * (x * s)), 2.0)) * cos((x * 2.0));
	} else {
		tmp = (pow(cos(x), 2.0) - pow(sin(x), 2.0)) / pow((s * (x * c)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 5d+47) then
        tmp = (1.0d0 / ((c * (x * s)) ** 2.0d0)) * cos((x * 2.0d0))
    else
        tmp = ((cos(x) ** 2.0d0) - (sin(x) ** 2.0d0)) / ((s * (x * c)) ** 2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e+47) {
		tmp = (1.0 / Math.pow((c * (x * s)), 2.0)) * Math.cos((x * 2.0));
	} else {
		tmp = (Math.pow(Math.cos(x), 2.0) - Math.pow(Math.sin(x), 2.0)) / Math.pow((s * (x * c)), 2.0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 5e+47:
		tmp = (1.0 / math.pow((c * (x * s)), 2.0)) * math.cos((x * 2.0))
	else:
		tmp = (math.pow(math.cos(x), 2.0) - math.pow(math.sin(x), 2.0)) / math.pow((s * (x * c)), 2.0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 5e+47)
		tmp = Float64(Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0)) * cos(Float64(x * 2.0)));
	else
		tmp = Float64(Float64((cos(x) ^ 2.0) - (sin(x) ^ 2.0)) / (Float64(s * Float64(x * c)) ^ 2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5e+47)
		tmp = (1.0 / ((c * (x * s)) ^ 2.0)) * cos((x * 2.0));
	else
		tmp = ((cos(x) ^ 2.0) - (sin(x) ^ 2.0)) / ((s * (x * c)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 5e+47], N[(N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000022e47

   1. Initial program 65.7%

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

      1. associate-/r* 65.8%

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

      2. *-commutative 65.8%

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

      3. associate-*l* 62.5%

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

      4. unpow2 62.5%

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

      5. unpow2 62.5%

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

      6. associate-*r* 68.7%

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

      7. associate-/r* 71.7%

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

      8. associate-/l/ 71.6%

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

      9. associate-/l/ 71.7%

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

      10. *-commutative 71.7%

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

      11. associate-*l* 67.7%

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

      12. unpow2 67.7%

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

      13. associate-*l* 62.1%

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

      14. unpow2 62.1%

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

      15. unswap-sqr 80.6%

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

      16. *-commutative 80.6%

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

      17. *-commutative 80.6%

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

   3. Simplified 80.6%

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

      1. clear-num 80.6%

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

      2. associate-/r/ 80.6%

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

   5. Applied egg-rr 97.5%

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

   if 5.00000000000000022e47 < 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* 66.0%

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

      2. remove-double-neg 66.0%

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

      3. distribute-lft-neg-out 66.0%

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

      4. distribute-lft-neg-out 66.0%

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

      5. distribute-rgt-neg-out 66.0%

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

      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* 69.0%

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

      10. distribute-lft-neg-in 69.0%

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

      11. distribute-lft-neg-out 69.0%

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

      12. remove-double-neg 69.0%

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

      13. associate-*r* 69.0%

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

      14. *-commutative 69.0%

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

      15. associate-*r* 67.4%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around 0 55.8%

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

      1. unpow2 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. swap-sqr 79.1%

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

      5. swap-sqr 94.9%

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

      6. unpow2 94.9%

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

      7. associate-*r* 97.2%

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

      8. *-commutative 97.2%

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

      9. associate-*l* 96.4%

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

   6. Simplified 96.4%

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

      1. cos-2 96.2%

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

      2. sub-neg 96.2%

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

      3. pow2 96.2%

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

      4. pow2 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. sub-neg 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\cos x}^{2} - {\sin x}^{2}}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \]

Alternative 2: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* x 2.0))))
   (if (<= x 5e+70)
     (* (/ 1.0 t_0) (/ t_1 t_0))
     (/ t_1 (* (* x c) (* s (* s (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 5e+70) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = t_1 / ((x * c) * (s * (s * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((x * 2.0d0))
    if (x <= 5d+70) then
        tmp = (1.0d0 / t_0) * (t_1 / t_0)
    else
        tmp = t_1 / ((x * c) * (s * (s * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 5e+70) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = t_1 / ((x * c) * (s * (s * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 5e+70:
		tmp = (1.0 / t_0) * (t_1 / t_0)
	else:
		tmp = t_1 / ((x * c) * (s * (s * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 5e+70)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / t_0));
	else
		tmp = Float64(t_1 / Float64(Float64(x * c) * Float64(s * Float64(s * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 5e+70)
		tmp = (1.0 / t_0) * (t_1 / t_0);
	else
		tmp = t_1 / ((x * c) * (s * (s * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5e+70], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e70

   1. Initial program 65.3%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-*l* 62.1%

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

      4. unpow2 62.1%

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

      5. unpow2 62.1%

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

      6. associate-*r* 68.2%

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

      7. associate-/r* 71.1%

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

      8. associate-/l/ 71.1%

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

      9. associate-/l/ 71.1%

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

      10. *-commutative 71.1%

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

      11. associate-*l* 67.3%

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

      12. unpow2 67.3%

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

      13. associate-*l* 61.6%

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

      14. unpow2 61.6%

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

      15. unswap-sqr 80.9%

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

      16. *-commutative 80.9%

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

      17. *-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. *-un-lft-identity 80.9%

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

      2. add-sqr-sqrt 80.9%

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

      3. times-frac 80.9%

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

   5. Applied egg-rr 97.4%

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

   if 5.0000000000000002e70 < x

   1. Initial program 68.0%

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

      1. associate-/r* 67.7%

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

      2. remove-double-neg 67.7%

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

      3. distribute-lft-neg-out 67.7%

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

      4. distribute-lft-neg-out 67.7%

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

      5. distribute-rgt-neg-out 67.7%

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

      6. associate-/l/ 68.0%

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

      7. distribute-rgt-neg-out 68.0%

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

      8. distribute-lft-neg-out 68.0%

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

      9. associate-*l* 72.5%

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

      10. distribute-lft-neg-in 72.5%

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

      11. distribute-lft-neg-out 72.5%

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

      12. remove-double-neg 72.5%

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

      13. associate-*r* 72.5%

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

      14. *-commutative 72.5%

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

      15. associate-*r* 69.2%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around 0 57.0%

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

      1. unpow2 57.0%

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

      2. unpow2 57.0%

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

      3. unpow2 57.0%

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

      4. swap-sqr 81.5%

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

      5. swap-sqr 94.7%

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

      6. unpow2 94.7%

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

      7. associate-*r* 97.2%

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

      8. *-commutative 97.2%

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

      9. associate-*l* 97.9%

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

   6. Simplified 97.9%

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

      1. unpow2 97.9%

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

      2. associate-*r* 95.6%

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

      3. *-commutative 95.6%

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

      4. associate-*r* 93.1%

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

      5. associate-*r* 89.9%

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

      6. associate-*r* 93.1%

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

      7. *-commutative 93.1%

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

      8. associate-*r* 94.7%

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

   8. Applied egg-rr 94.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{-26}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\cos \left(x \cdot 2\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 1.5e-26)
     (pow (* c (* x s)) -2.0)
     (* (/ 1.0 t_0) (/ (cos (* x 2.0)) t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.5e-26) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (1.0 / t_0) * (cos((x * 2.0)) / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 1.5d-26) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (1.0d0 / t_0) * (cos((x * 2.0d0)) / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.5e-26) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (1.0 / t_0) * (Math.cos((x * 2.0)) / t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 1.5e-26:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (1.0 / t_0) * (math.cos((x * 2.0)) / t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 1.5e-26)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x * 2.0)) / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 1.5e-26)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (1.0 / t_0) * (cos((x * 2.0)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.5e-26], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $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. Split input into 2 regimes

2. if x < 1.50000000000000006e-26

   1. Initial program 65.6%

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

      1. associate-/r* 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-lft-neg-out 65.6%

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

      4. distribute-lft-neg-out 65.6%

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

      5. distribute-rgt-neg-out 65.6%

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

      6. associate-/l/ 65.6%

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

      7. distribute-rgt-neg-out 65.6%

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

      8. distribute-lft-neg-out 65.6%

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

      9. associate-*l* 67.0%

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

      10. distribute-lft-neg-in 67.0%

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

      11. distribute-lft-neg-out 67.0%

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

      12. remove-double-neg 67.0%

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

      13. associate-*r* 66.7%

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

      14. *-commutative 66.7%

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

      15. associate-*r* 64.7%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around 0 62.2%

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

      1. unpow2 62.2%

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

      2. unpow2 62.2%

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

      3. unpow2 62.2%

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

      4. swap-sqr 74.9%

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

      5. swap-sqr 97.4%

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

      6. unpow2 97.4%

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

      7. associate-*r* 98.1%

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

      8. *-commutative 98.1%

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

      9. associate-*l* 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in x around 0 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. unpow2 56.5%

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

      4. associate-*r* 56.5%

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

      5. *-commutative 56.5%

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

      6. swap-sqr 70.6%

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

      7. swap-sqr 85.2%

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

      8. associate-*r* 83.7%

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

      9. associate-*r* 84.1%

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

      10. unpow2 84.1%

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

      11. /-rgt-identity 84.1%

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

      12. unpow2 84.1%

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

      13. associate-/l* 84.1%

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

      14. associate-/l* 84.2%

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

      15. associate-*l/ 84.1%

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

      16. unpow-1 84.1%

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

      17. unpow-1 84.1%

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

      18. pow-sqr 84.2%

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

   9. Simplified 84.4%

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

   if 1.50000000000000006e-26 < x

   1. Initial program 66.6%

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

      1. associate-/r* 66.3%

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

      2. remove-double-neg 66.3%

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

      3. distribute-lft-neg-out 66.3%

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

      4. distribute-lft-neg-out 66.3%

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

      5. distribute-rgt-neg-out 66.3%

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

      6. associate-/l/ 66.6%

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

      7. distribute-rgt-neg-out 66.6%

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

      8. distribute-lft-neg-out 66.6%

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

      9. associate-*l* 68.9%

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

      10. distribute-lft-neg-in 68.9%

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

      11. distribute-lft-neg-out 68.9%

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

      12. remove-double-neg 68.9%

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

      13. associate-*r* 68.9%

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

      14. *-commutative 68.9%

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

      15. associate-*r* 67.5%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in x around 0 57.4%

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

      1. unpow2 57.4%

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

      2. unpow2 57.4%

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

      3. unpow2 57.4%

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

      4. swap-sqr 77.7%

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

      5. swap-sqr 95.5%

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

      6. unpow2 95.5%

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

      7. associate-*r* 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-*l* 96.8%

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

   6. Simplified 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. associate-*r* 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*r* 95.5%

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

      5. unpow2 95.5%

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

      6. times-frac 95.6%

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

      7. associate-*r* 93.7%

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

      8. *-commutative 93.7%

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

      9. associate-*r* 93.0%

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

      10. *-commutative 93.0%

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

      11. associate-*r* 95.1%

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

      12. *-commutative 95.1%

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

      13. associate-*r* 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-26}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 4: 88.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.016)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* x (* x (* c (* c (* s s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.016) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.016d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (c * (s * s)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.016) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.016:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.016)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.016)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.016], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $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 < 0.016

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. remove-double-neg 66.2%

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

      3. distribute-lft-neg-out 66.2%

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

      4. distribute-lft-neg-out 66.2%

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

      5. distribute-rgt-neg-out 66.2%

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

      6. associate-/l/ 66.2%

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

      7. distribute-rgt-neg-out 66.2%

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

      8. distribute-lft-neg-out 66.2%

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

      9. associate-*l* 67.6%

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

      10. distribute-lft-neg-in 67.6%

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

      11. distribute-lft-neg-out 67.6%

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

      12. remove-double-neg 67.6%

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

      13. associate-*r* 67.2%

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

      14. *-commutative 67.2%

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

      15. associate-*r* 65.3%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 75.3%

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

      5. swap-sqr 97.4%

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

      6. unpow2 97.4%

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

      7. associate-*r* 98.1%

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

      8. *-commutative 98.1%

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

      9. associate-*l* 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. associate-*r* 57.2%

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

      5. *-commutative 57.2%

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

      6. swap-sqr 71.0%

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

      7. swap-sqr 85.4%

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

      8. associate-*r* 83.9%

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

      9. associate-*r* 84.4%

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

      10. unpow2 84.4%

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

      11. /-rgt-identity 84.4%

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

      12. unpow2 84.4%

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

      13. associate-/l* 84.4%

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

      14. associate-/l* 84.4%

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

      15. associate-*l/ 84.3%

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

      16. unpow-1 84.3%

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

      17. unpow-1 84.3%

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

      18. pow-sqr 84.4%

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

   9. Simplified 84.7%

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

   if 0.016 < x

   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* 64.8%

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

      2. remove-double-neg 64.8%

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

      3. distribute-lft-neg-out 64.8%

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

      4. distribute-lft-neg-out 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. associate-/l/ 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* 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 73.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 95.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.0065)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* x (* (* x c) (* s (* c s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0065) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (x * ((x * c) * (s * (c * s))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.0065d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (x * ((x * c) * (s * (c * s))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0065) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * ((x * c) * (s * (c * s))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.0065:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (x * ((x * c) * (s * (c * s))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.0065)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(x * c) * Float64(s * Float64(c * s)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.0065)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (x * ((x * c) * (s * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.0065], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(x * c), $MachinePrecision] * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0064999999999999997

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. remove-double-neg 66.2%

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

      3. distribute-lft-neg-out 66.2%

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

      4. distribute-lft-neg-out 66.2%

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

      5. distribute-rgt-neg-out 66.2%

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

      6. associate-/l/ 66.2%

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

      7. distribute-rgt-neg-out 66.2%

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

      8. distribute-lft-neg-out 66.2%

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

      9. associate-*l* 67.6%

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

      10. distribute-lft-neg-in 67.6%

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

      11. distribute-lft-neg-out 67.6%

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

      12. remove-double-neg 67.6%

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

      13. associate-*r* 67.2%

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

      14. *-commutative 67.2%

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

      15. associate-*r* 65.3%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 75.3%

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

      5. swap-sqr 97.4%

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

      6. unpow2 97.4%

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

      7. associate-*r* 98.1%

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

      8. *-commutative 98.1%

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

      9. associate-*l* 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. associate-*r* 57.2%

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

      5. *-commutative 57.2%

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

      6. swap-sqr 71.0%

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

      7. swap-sqr 85.4%

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

      8. associate-*r* 83.9%

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

      9. associate-*r* 84.4%

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

      10. unpow2 84.4%

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

      11. /-rgt-identity 84.4%

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

      12. unpow2 84.4%

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

      13. associate-/l* 84.4%

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

      14. associate-/l* 84.4%

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

      15. associate-*l/ 84.3%

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

      16. unpow-1 84.3%

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

      17. unpow-1 84.3%

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

      18. pow-sqr 84.4%

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

   9. Simplified 84.7%

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

   if 0.0064999999999999997 < x

   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* 64.8%

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

      2. remove-double-neg 64.8%

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

      3. distribute-lft-neg-out 64.8%

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

      4. distribute-lft-neg-out 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. associate-/l/ 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* 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 73.7%

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

      1. add-sqr-sqrt 73.7%

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

      2. pow2 73.7%

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

      3. associate-*r* 66.1%

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

      4. swap-sqr 83.9%

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

      5. *-commutative 83.9%

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

      6. sqrt-prod 83.7%

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

      7. sqrt-prod 54.6%

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

      8. add-sqr-sqrt 92.4%

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

   5. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. *-commutative 92.4%

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

      3. *-commutative 92.4%

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

      4. swap-sqr 83.8%

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

      5. add-sqr-sqrt 83.9%

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

      6. associate-*l* 82.4%

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

      7. associate-*r* 82.5%

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

      8. *-commutative 82.5%

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

      9. *-commutative 82.5%

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

   7. Applied egg-rr 82.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 6: 96.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-37}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4e-37)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* x (* (* s (* x c)) (* c s))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e-37) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (x * ((s * (x * c)) * (c * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4d-37) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (x * ((s * (x * c)) * (c * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e-37) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * ((s * (x * c)) * (c * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4e-37:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (x * ((s * (x * c)) * (c * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4e-37)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(s * Float64(x * c)) * Float64(c * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4e-37)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (x * ((s * (x * c)) * (c * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4e-37], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000027e-37

   1. Initial program 65.6%

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

      1. associate-/r* 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-lft-neg-out 65.6%

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

      4. distribute-lft-neg-out 65.6%

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

      5. distribute-rgt-neg-out 65.6%

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

      6. associate-/l/ 65.6%

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

      7. distribute-rgt-neg-out 65.6%

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

      8. distribute-lft-neg-out 65.6%

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

      9. associate-*l* 67.0%

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

      10. distribute-lft-neg-in 67.0%

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

      11. distribute-lft-neg-out 67.0%

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

      12. remove-double-neg 67.0%

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

      13. associate-*r* 66.6%

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

      14. *-commutative 66.6%

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

      15. associate-*r* 64.6%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. unpow2 62.1%

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

      4. swap-sqr 74.9%

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

      5. swap-sqr 97.3%

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

      6. unpow2 97.3%

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

      7. associate-*r* 98.0%

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

      8. *-commutative 98.0%

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

      9. associate-*l* 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in x around 0 56.3%

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

      1. unpow2 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

      4. associate-*r* 56.3%

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

      5. *-commutative 56.3%

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

      6. swap-sqr 70.6%

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

      7. swap-sqr 84.9%

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

      8. associate-*r* 83.4%

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

      9. associate-*r* 83.9%

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

      10. unpow2 83.9%

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

      11. /-rgt-identity 83.9%

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

      12. unpow2 83.9%

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

      13. associate-/l* 83.8%

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

      14. associate-/l* 83.9%

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

      15. associate-*l/ 83.8%

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

      16. unpow-1 83.8%

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

      17. unpow-1 83.8%

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

      18. pow-sqr 83.9%

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

   9. Simplified 84.2%

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

   if 4.00000000000000027e-37 < x

   1. Initial program 66.7%

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

      1. associate-/r* 66.4%

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

      2. remove-double-neg 66.4%

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

      3. distribute-lft-neg-out 66.4%

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

      4. distribute-lft-neg-out 66.4%

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

      5. distribute-rgt-neg-out 66.4%

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

      6. associate-/l/ 66.7%

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

      7. distribute-rgt-neg-out 66.7%

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

      8. distribute-lft-neg-out 66.7%

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

      9. associate-*l* 68.9%

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

      10. distribute-lft-neg-in 68.9%

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

      11. distribute-lft-neg-out 68.9%

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

      12. remove-double-neg 68.9%

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

      13. associate-*r* 69.0%

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

      14. *-commutative 69.0%

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

      15. associate-*r* 67.6%

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

   3. Simplified 74.6%

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

      1. add-sqr-sqrt 74.6%

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

      2. pow2 74.6%

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

      3. associate-*r* 67.6%

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

      4. swap-sqr 83.9%

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

      5. *-commutative 83.9%

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

      6. sqrt-prod 83.8%

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

      7. sqrt-prod 55.5%

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

      8. add-sqr-sqrt 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. unpow2 93.0%

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

      2. *-commutative 93.0%

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

      3. associate-*r* 93.0%

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

      4. associate-*r* 93.1%

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

      5. add-sqr-sqrt 93.1%

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

      6. *-commutative 93.1%

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

      7. associate-*r* 90.6%

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

      8. *-commutative 90.6%

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

   7. Applied egg-rr 90.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-37}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]

Alternative 7: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.0065)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* (* x c) (* s (* s (* x c)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0065) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / ((x * c) * (s * (s * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.0065d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / ((x * c) * (s * (s * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0065) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / ((x * c) * (s * (s * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.0065:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / ((x * c) * (s * (s * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.0065)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(Float64(x * c) * Float64(s * Float64(s * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.0065)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / ((x * c) * (s * (s * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.0065], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x * c), $MachinePrecision] * N[(s * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0064999999999999997

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. remove-double-neg 66.2%

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

      3. distribute-lft-neg-out 66.2%

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

      4. distribute-lft-neg-out 66.2%

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

      5. distribute-rgt-neg-out 66.2%

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

      6. associate-/l/ 66.2%

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

      7. distribute-rgt-neg-out 66.2%

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

      8. distribute-lft-neg-out 66.2%

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

      9. associate-*l* 67.6%

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

      10. distribute-lft-neg-in 67.6%

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

      11. distribute-lft-neg-out 67.6%

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

      12. remove-double-neg 67.6%

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

      13. associate-*r* 67.2%

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

      14. *-commutative 67.2%

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

      15. associate-*r* 65.3%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 75.3%

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

      5. swap-sqr 97.4%

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

      6. unpow2 97.4%

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

      7. associate-*r* 98.1%

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

      8. *-commutative 98.1%

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

      9. associate-*l* 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. associate-*r* 57.2%

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

      5. *-commutative 57.2%

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

      6. swap-sqr 71.0%

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

      7. swap-sqr 85.4%

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

      8. associate-*r* 83.9%

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

      9. associate-*r* 84.4%

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

      10. unpow2 84.4%

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

      11. /-rgt-identity 84.4%

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

      12. unpow2 84.4%

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

      13. associate-/l* 84.4%

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

      14. associate-/l* 84.4%

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

      15. associate-*l/ 84.3%

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

      16. unpow-1 84.3%

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

      17. unpow-1 84.3%

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

      18. pow-sqr 84.4%

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

   9. Simplified 84.7%

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

   if 0.0064999999999999997 < x

   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* 64.8%

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

      2. remove-double-neg 64.8%

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

      3. distribute-lft-neg-out 64.8%

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

      4. distribute-lft-neg-out 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. associate-/l/ 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* 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 73.7%

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

   4. Taylor expanded in x around 0 55.5%

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

      1. unpow2 55.5%

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

      2. unpow2 55.5%

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

      3. unpow2 55.5%

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

      4. swap-sqr 76.7%

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

      5. swap-sqr 95.3%

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

      6. unpow2 95.3%

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

      7. associate-*r* 97.4%

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

      8. *-commutative 97.4%

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

      9. associate-*l* 96.6%

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

   6. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. associate-*r* 94.7%

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

      3. *-commutative 94.7%

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

      4. associate-*r* 92.5%

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

      5. associate-*r* 89.7%

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

      6. associate-*r* 92.5%

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

      7. *-commutative 92.5%

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

      8. associate-*r* 93.9%

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

   8. Applied egg-rr 93.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 8: 79.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

4. Taylor expanded in x around 0 63.4%

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

   1. associate-*r* 58.2%

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

   2. *-commutative 58.2%

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

   3. associate-*r* 58.6%

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

   4. associate-*l* 58.2%

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

   5. *-commutative 58.2%

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

   6. add-sqr-sqrt 58.2%

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

   7. pow2 58.2%

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

   8. sqrt-prod 58.2%

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

   9. sqrt-prod 31.4%

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

   10. add-sqr-sqrt 62.9%

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

   11. sqrt-prod 35.0%

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

   12. sqrt-prod 35.7%

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

   13. sqrt-prod 24.1%

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

   14. add-sqr-sqrt 41.5%

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

   15. associate-*l* 41.5%

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

   16. add-sqr-sqrt 78.3%

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

   17. *-commutative 78.3%

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

6. Applied egg-rr 78.3%

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

7. Final simplification 78.3%

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

Alternative 9: 79.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

4. Taylor expanded in x around 0 60.9%

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

   1. unpow2 60.9%

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

   2. unpow2 60.9%

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

   3. unpow2 60.9%

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

   4. swap-sqr 75.6%

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

   5. swap-sqr 96.9%

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

   6. unpow2 96.9%

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

   7. associate-*r* 97.9%

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

   8. *-commutative 97.9%

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

   9. associate-*l* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 55.7%

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

   1. unpow2 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. associate-*r* 55.6%

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

   5. *-commutative 55.6%

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

   6. swap-sqr 67.0%

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

   7. swap-sqr 78.8%

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

   8. associate-*r* 77.6%

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

   9. associate-*r* 78.0%

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

   10. unpow2 78.0%

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

   11. /-rgt-identity 78.0%

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

   12. unpow2 78.0%

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

   13. associate-/l* 78.0%

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

   14. associate-/l* 78.0%

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

   15. associate-*l/ 77.9%

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

   16. unpow-1 77.9%

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

   17. unpow-1 77.9%

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

   18. pow-sqr 78.0%

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

9. Simplified 78.3%

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

10. Final simplification 78.3%

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

Alternative 10: 79.6% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

4. Taylor expanded in x around 0 60.9%

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

   1. unpow2 60.9%

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

   2. unpow2 60.9%

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

   3. unpow2 60.9%

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

   4. swap-sqr 75.6%

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

   5. swap-sqr 96.9%

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

   6. unpow2 96.9%

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

   7. associate-*r* 97.9%

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

   8. *-commutative 97.9%

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

   9. associate-*l* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 55.7%

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

   1. unpow2 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. associate-*r* 55.6%

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

   5. *-commutative 55.6%

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

   6. swap-sqr 67.0%

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

   7. swap-sqr 78.8%

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

   8. associate-*r* 77.6%

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

   9. associate-*r* 78.0%

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

   10. unpow2 78.0%

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

   11. /-rgt-identity 78.0%

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

   12. unpow2 78.0%

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

   13. associate-/l* 78.0%

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

   14. associate-/l* 78.0%

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

   15. associate-*l/ 77.9%

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

   16. unpow-1 77.9%

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

   17. unpow-1 77.9%

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

   18. pow-sqr 78.0%

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

9. Simplified 78.3%

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

    1. metadata-eval 78.3%

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

    2. pow-prod-up 78.2%

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

    3. unpow-1 78.2%

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

    4. *-commutative 78.2%

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

    5. unpow-1 78.2%

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

    6. *-commutative 78.2%

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

11. Applied egg-rr 78.2%

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

12. Final simplification 78.2%

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

Alternative 11: 63.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

4. Taylor expanded in x around 0 63.4%

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

5. Final simplification 63.4%

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

Alternative 12: 74.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

   1. add-sqr-sqrt 37.3%

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

   2. pow2 37.3%

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

   3. associate-*r* 33.9%

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

   4. swap-sqr 41.9%

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

   5. *-commutative 41.9%

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

   6. sqrt-prod 41.9%

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

   7. sqrt-prod 28.8%

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

   8. add-sqr-sqrt 48.2%

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

5. Applied egg-rr 48.2%

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

   1. unpow2 48.2%

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

   2. *-commutative 48.2%

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

   3. associate-*r* 48.2%

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

   4. associate-*r* 48.2%

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

   5. add-sqr-sqrt 93.8%

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

   6. *-commutative 93.8%

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

   7. associate-*r* 91.7%

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

   8. *-commutative 91.7%

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

7. Applied egg-rr 91.7%

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

8. Taylor expanded in x around 0 74.6%

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

9. Final simplification 74.6%

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

Alternative 13: 74.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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/ 65.9%

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

   7. distribute-rgt-neg-out 65.9%

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

   8. distribute-lft-neg-out 65.9%

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

   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* 67.3%

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

   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* 65.5%

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

3. Simplified 73.1%

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

4. Taylor expanded in x around 0 60.9%

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

   1. unpow2 60.9%

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

   2. unpow2 60.9%

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

   3. unpow2 60.9%

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

   4. swap-sqr 75.6%

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

   5. swap-sqr 96.9%

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

   6. unpow2 96.9%

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

   7. associate-*r* 97.9%

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

   8. *-commutative 97.9%

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

   9. associate-*l* 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

   2. associate-*r* 95.8%

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

   3. *-commutative 95.8%

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

   4. associate-*r* 94.7%

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

   5. associate-*r* 89.7%

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

   6. associate-*r* 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-*r* 91.7%

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

8. Applied egg-rr 91.7%

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

9. Taylor expanded in x around 0 74.5%

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

10. Final simplification 74.5%

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

Alternative 14: 27.7% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. *-commutative 65.8%

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

   3. associate-*l* 60.9%

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

   4. unpow2 60.9%

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

   5. unpow2 60.9%

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

   6. associate-*r* 66.8%

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

   7. associate-/r* 69.0%

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

   8. associate-/l/ 69.1%

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

   9. associate-/l/ 69.1%

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

   10. *-commutative 69.1%

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

   11. associate-*l* 66.0%

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

   12. unpow2 66.0%

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

   13. associate-*l* 60.8%

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

   14. unpow2 60.8%

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

   15. unswap-sqr 77.8%

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

   16. *-commutative 77.8%

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

   17. *-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in x around 0 49.8%

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

   1. unpow2 49.8%

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

6. Simplified 49.8%

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

7. Taylor expanded in x around inf 27.2%

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

   1. unpow2 27.2%

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

   2. unpow2 27.2%

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

   3. associate-/r* 27.2%

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

9. Simplified 27.2%

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

10. Taylor expanded in c around 0 27.2%

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

    1. unpow2 27.2%

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

    2. unpow2 27.2%

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

12. Simplified 27.2%

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

13. Final simplification 27.2%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))