mixedcos

Percentage Accurate: 66.3% → 99.4%

Time: 12.9s

Alternatives: 11

Speedup: 24.1×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-75}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

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))) (t_1 (cos (* x 2.0))))
   (if (<= x 5e-75)
     (* (pow (* c (* x s)) -2.0) t_1)
     (* (/ t_1 t_0) (/ 1.0 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 t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 5e-75) {
		tmp = pow((c * (x * s)), -2.0) * t_1;
	} else {
		tmp = (t_1 / t_0) * (1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((x * 2.0d0))
    if (x <= 5d-75) then
        tmp = ((c * (x * s)) ** (-2.0d0)) * t_1
    else
        tmp = (t_1 / t_0) * (1.0d0 / 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 t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 5e-75) {
		tmp = Math.pow((c * (x * s)), -2.0) * t_1;
	} else {
		tmp = (t_1 / t_0) * (1.0 / 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)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 5e-75:
		tmp = math.pow((c * (x * s)), -2.0) * t_1
	else:
		tmp = (t_1 / t_0) * (1.0 / 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))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 5e-75)
		tmp = Float64((Float64(c * Float64(x * s)) ^ -2.0) * t_1);
	else
		tmp = Float64(Float64(t_1 / t_0) * Float64(1.0 / 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);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 5e-75)
		tmp = ((c * (x * s)) ^ -2.0) * t_1;
	else
		tmp = (t_1 / t_0) * (1.0 / 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]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5e-75], N[(N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999979e-75

   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. *-commutative 65.9%

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

      2. associate-*r* 60.7%

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

      3. associate-*r* 60.7%

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

      4. unpow2 60.7%

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

      5. unswap-sqr 74.6%

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

      6. unpow2 74.6%

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

      7. swap-sqr 93.9%

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

      8. *-commutative 93.9%

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

      9. *-commutative 93.9%

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

      10. *-commutative 93.9%

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

      11. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in x around inf 60.7%

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

      1. associate-/r* 60.5%

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

      2. *-commutative 60.5%

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

      3. unpow2 60.5%

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

      4. unpow2 60.5%

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

      5. swap-sqr 77.9%

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

      6. unpow2 77.9%

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

      7. associate-/l/ 78.1%

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

      8. unpow2 78.1%

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

      9. unpow2 78.1%

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

      10. swap-sqr 96.8%

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

      11. associate-*r* 91.7%

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

      12. associate-*r* 93.9%

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

      13. associate-/r* 94.2%

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

      14. *-rgt-identity 94.2%

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

      15. associate-*r/ 94.2%

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

   6. Simplified 97.2%

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

   if 4.99999999999999979e-75 < x

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*r* 55.6%

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

      3. associate-*r* 54.6%

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

      4. unpow2 54.6%

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

      5. unswap-sqr 66.0%

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

      6. unpow2 66.0%

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

      7. swap-sqr 97.9%

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

      8. *-commutative 97.9%

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

      9. *-commutative 97.9%

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

      10. *-commutative 97.9%

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

      11. *-commutative 97.9%

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

   3. Simplified 97.9%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.4%

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

      3. *-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-75}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 2: 86.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot {\left(x \cdot c\right)}^{2}\right)}\\ \end{array} \end{array} \]

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 (/ (/ 1.0 c) (* x s))))
   (if (<= x 9e-9)
     (* t_0 t_0)
     (if (<= x 6.6e+150)
       (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
       (/ 1.0 (* s (* s (pow (* 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 t_0 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 9e-9) {
		tmp = t_0 * t_0;
	} else if (x <= 6.6e+150) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 1.0 / (s * (s * pow((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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x * s)
    if (x <= 9d-9) then
        tmp = t_0 * t_0
    else if (x <= 6.6d+150) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = 1.0d0 / (s * (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 t_0 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 9e-9) {
		tmp = t_0 * t_0;
	} else if (x <= 6.6e+150) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 1.0 / (s * (s * Math.pow((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):
	t_0 = (1.0 / c) / (x * s)
	tmp = 0
	if x <= 9e-9:
		tmp = t_0 * t_0
	elif x <= 6.6e+150:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = 1.0 / (s * (s * math.pow((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)
	t_0 = Float64(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (x <= 9e-9)
		tmp = Float64(t_0 * t_0);
	elseif (x <= 6.6e+150)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = Float64(1.0 / Float64(s * 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)
	t_0 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (x <= 9e-9)
		tmp = t_0 * t_0;
	elseif (x <= 6.6e+150)
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = 1.0 / (s * (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_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9e-9], N[(t$95$0 * t$95$0), $MachinePrecision], If[LessEqual[x, 6.6e+150], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(s * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(s * N[Power[N[(x * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.99999999999999953e-9

   1. Initial program 66.1%

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

      1. associate-/r* 65.8%

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

      2. unpow2 65.8%

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

      3. *-commutative 65.8%

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

      4. unpow2 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in x around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

      1. add-sqr-sqrt 60.7%

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

      2. sqrt-div 60.7%

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

      3. sqrt-div 60.7%

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

      4. metadata-eval 60.7%

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

      5. sqrt-prod 27.7%

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

      6. add-sqr-sqrt 39.0%

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

      7. associate-*r* 36.9%

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

      8. sqrt-prod 36.9%

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

      9. sqrt-unprod 12.6%

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

      10. add-sqr-sqrt 44.3%

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

      11. sqrt-prod 24.4%

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

      12. add-sqr-sqrt 44.1%

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

      13. *-commutative 44.1%

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

      14. sqrt-div 44.0%

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

      15. sqrt-div 44.6%

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

      16. metadata-eval 44.6%

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

      17. sqrt-prod 22.6%

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

      18. add-sqr-sqrt 42.7%

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

      19. associate-*r* 40.2%

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

      20. sqrt-prod 41.8%

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

   8. Applied egg-rr 84.7%

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

   if 8.99999999999999953e-9 < x < 6.59999999999999962e150

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 61.6%

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

      3. associate-*r* 61.9%

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

      4. *-commutative 61.9%

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

      5. unpow2 61.9%

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

      6. associate-*r* 75.2%

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

      7. associate-*r* 77.4%

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

      8. *-commutative 77.4%

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

      9. unpow2 77.4%

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

   3. Simplified 77.4%

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

   if 6.59999999999999962e150 < x

   1. Initial program 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r* 43.7%

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

      3. associate-*r* 43.7%

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

      4. unpow2 43.7%

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

      5. unswap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. swap-sqr 95.6%

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

      8. *-commutative 95.6%

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

      9. *-commutative 95.6%

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

      10. *-commutative 95.6%

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

      11. *-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 43.7%

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

      1. associate-/r* 43.7%

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

      2. unpow2 43.7%

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

      3. unpow2 43.7%

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

      4. swap-sqr 60.7%

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

      5. unpow2 60.7%

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

      6. associate-/l/ 60.7%

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

      7. unpow2 60.7%

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

      8. unpow2 60.7%

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

      9. swap-sqr 71.1%

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

      10. associate-*r* 70.8%

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

      11. associate-*r* 70.8%

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

      12. unpow2 70.8%

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

      13. associate-*r* 71.1%

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

      14. *-commutative 71.1%

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

   6. Simplified 71.1%

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

      1. associate-*r* 70.9%

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

      2. pow2 70.9%

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

      3. *-commutative 70.9%

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

      4. associate-*r* 70.6%

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

      5. associate-*r* 69.5%

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

      6. *-commutative 69.5%

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

      7. *-commutative 69.5%

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

      8. associate-*r* 69.7%

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

      9. associate-*r* 69.7%

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

      10. pow2 69.7%

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

      11. *-commutative 69.7%

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

   8. Applied egg-rr 69.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot {\left(x \cdot c\right)}^{2}\right)}\\ \end{array} \]

Alternative 3: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;t_1 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

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))) (t_1 (/ (/ 1.0 c) (* x s))))
   (if (<= x 1.3e-77) (* t_1 t_1) (* (/ (cos (* x 2.0)) t_0) (/ 1.0 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 t_1 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 1.3e-77) {
		tmp = t_1 * t_1;
	} else {
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = (1.0d0 / c) / (x * s)
    if (x <= 1.3d-77) then
        tmp = t_1 * t_1
    else
        tmp = (cos((x * 2.0d0)) / t_0) * (1.0d0 / 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 t_1 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 1.3e-77) {
		tmp = t_1 * t_1;
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) * (1.0 / 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)
	t_1 = (1.0 / c) / (x * s)
	tmp = 0
	if x <= 1.3e-77:
		tmp = t_1 * t_1
	else:
		tmp = (math.cos((x * 2.0)) / t_0) * (1.0 / 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))
	t_1 = Float64(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (x <= 1.3e-77)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) * Float64(1.0 / 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);
	t_1 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (x <= 1.3e-77)
		tmp = t_1 * t_1;
	else
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / 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]}, Block[{t$95$1 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.3e-77], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e-77

   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* 65.9%

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

      2. unpow2 65.9%

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

      3. *-commutative 65.9%

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

      4. unpow2 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in x around 0 60.2%

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

      1. unpow2 60.2%

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

   6. Simplified 60.2%

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

      1. add-sqr-sqrt 60.2%

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

      2. sqrt-div 60.2%

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

      3. sqrt-div 60.2%

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

      4. metadata-eval 60.2%

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

      5. sqrt-prod 28.7%

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

      6. add-sqr-sqrt 39.9%

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

      7. associate-*r* 37.5%

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

      8. sqrt-prod 37.6%

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

      9. sqrt-unprod 10.6%

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

      10. add-sqr-sqrt 45.8%

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

      11. sqrt-prod 25.0%

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

      12. add-sqr-sqrt 45.5%

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

      13. *-commutative 45.5%

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

      14. sqrt-div 45.5%

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

      15. sqrt-div 46.1%

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

      16. metadata-eval 46.1%

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

      17. sqrt-prod 23.1%

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

      18. add-sqr-sqrt 41.5%

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

      19. associate-*r* 38.8%

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

      20. sqrt-prod 40.5%

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

   8. Applied egg-rr 83.2%

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

   if 1.3000000000000001e-77 < x

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*r* 55.4%

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

      3. associate-*r* 54.5%

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

      4. unpow2 54.5%

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

      5. unswap-sqr 65.6%

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

      6. unpow2 65.6%

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

      7. swap-sqr 97.9%

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

      8. *-commutative 97.9%

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

      9. *-commutative 97.9%

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

      10. *-commutative 97.9%

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

      11. *-commutative 97.9%

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

   3. Simplified 97.9%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.5%

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

      3. *-commutative 98.5%

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

   5. Applied egg-rr 98.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 4: 86.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;s \leq 1.95 \cdot 10^{+125}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \end{array} \]

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))))
   (if (<= s 1.95e+125)
     (/ (cos (* x 2.0)) (* x (* x (* (* s s) (* c c)))))
     (* 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);
	double tmp;
	if (s <= 1.95e+125) {
		tmp = cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x * s)
    if (s <= 1.95d+125) then
        tmp = cos((x * 2.0d0)) / (x * (x * ((s * s) * (c * c))))
    else
        tmp = t_0 * t_0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = (1.0 / c) / (x * s);
	double tmp;
	if (s <= 1.95e+125) {
		tmp = Math.cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = (1.0 / c) / (x * s)
	tmp = 0
	if s <= 1.95e+125:
		tmp = math.cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))))
	else:
		tmp = t_0 * t_0
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (s <= 1.95e+125)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(Float64(s * s) * Float64(c * c)))));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (s <= 1.95e+125)
		tmp = cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 1.95e+125], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.9500000000000001e125

   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* 69.2%

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

      2. *-commutative 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-*r* 67.5%

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

      5. *-commutative 67.5%

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

      6. unpow2 67.5%

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

      7. unpow2 67.5%

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

   3. Simplified 67.5%

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

   if 1.9500000000000001e125 < s

   1. Initial program 47.5%

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

      1. associate-/r* 46.5%

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

      2. unpow2 46.5%

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

      3. *-commutative 46.5%

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

      4. unpow2 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in x around 0 46.5%

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

      1. unpow2 46.5%

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

   6. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. sqrt-div 46.5%

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

      3. sqrt-div 46.6%

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

      4. metadata-eval 46.6%

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

      5. sqrt-prod 25.4%

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

      6. add-sqr-sqrt 42.1%

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

      7. associate-*r* 39.3%

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

      8. sqrt-prod 39.4%

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

      9. sqrt-unprod 25.8%

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

      10. add-sqr-sqrt 42.1%

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

      11. sqrt-prod 42.1%

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

      12. add-sqr-sqrt 42.1%

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

      13. *-commutative 42.1%

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

      14. sqrt-div 42.1%

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

      15. sqrt-div 44.5%

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

      16. metadata-eval 44.5%

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

      17. sqrt-prod 26.8%

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

      18. add-sqr-sqrt 48.5%

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

      19. associate-*r* 45.7%

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

      20. sqrt-prod 45.7%

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

   8. Applied egg-rr 86.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.95 \cdot 10^{+125}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \end{array} \]

Alternative 5: 90.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;s \leq 2.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \end{array} \]

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))))
   (if (<= s 2.05e+148)
     (/ (cos (* x 2.0)) (* x (* (* c (* x c)) (* s 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);
	double tmp;
	if (s <= 2.05e+148) {
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x * s)
    if (s <= 2.05d+148) then
        tmp = cos((x * 2.0d0)) / (x * ((c * (x * c)) * (s * s)))
    else
        tmp = t_0 * t_0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = (1.0 / c) / (x * s);
	double tmp;
	if (s <= 2.05e+148) {
		tmp = Math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = (1.0 / c) / (x * s)
	tmp = 0
	if s <= 2.05e+148:
		tmp = math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)))
	else:
		tmp = t_0 * t_0
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (s <= 2.05e+148)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (s <= 2.05e+148)
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 2.05e+148], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 2.0499999999999999e148

   1. Initial program 67.2%

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

      1. associate-*r* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r* 70.0%

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

      4. unpow2 70.0%

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

      5. unpow2 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in c around 0 70.0%

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

      1. unpow2 70.0%

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

      2. associate-*l* 75.4%

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

   6. Simplified 75.4%

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

   if 2.0499999999999999e148 < s

   1. Initial program 41.9%

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

      1. associate-/r* 40.7%

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

      2. unpow2 40.7%

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

      3. *-commutative 40.7%

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

      4. unpow2 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in x around 0 40.7%

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

      1. unpow2 40.7%

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

   6. Simplified 40.7%

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

      1. add-sqr-sqrt 40.7%

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

      2. sqrt-div 40.7%

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

      3. sqrt-div 40.7%

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

      4. metadata-eval 40.7%

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

      5. sqrt-prod 24.0%

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

      6. add-sqr-sqrt 40.7%

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

      7. associate-*r* 37.5%

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

      8. sqrt-prod 37.5%

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

      9. sqrt-unprod 29.8%

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

      10. add-sqr-sqrt 40.7%

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

      11. sqrt-prod 40.7%

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

      12. add-sqr-sqrt 40.7%

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

      13. *-commutative 40.7%

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

      14. sqrt-div 40.7%

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

      15. sqrt-div 43.4%

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

      16. metadata-eval 43.4%

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

      17. sqrt-prod 28.4%

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

      18. add-sqr-sqrt 50.8%

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

      19. associate-*r* 47.5%

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

      20. sqrt-prod 47.5%

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

   8. Applied egg-rr 84.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \end{array} \]

Alternative 6: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-60}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x s))))
   (if (<= x 4.5e-60)
     (* t_0 t_0)
     (/ (cos (* x 2.0)) (* (* c (* x 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 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 4.5e-60) {
		tmp = t_0 * t_0;
	} else {
		tmp = cos((x * 2.0)) / ((c * (x * 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) :: tmp
    t_0 = (1.0d0 / c) / (x * s)
    if (x <= 4.5d-60) then
        tmp = t_0 * t_0
    else
        tmp = cos((x * 2.0d0)) / ((c * (x * 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 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 4.5e-60) {
		tmp = t_0 * t_0;
	} else {
		tmp = Math.cos((x * 2.0)) / ((c * (x * 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 = (1.0 / c) / (x * s)
	tmp = 0
	if x <= 4.5e-60:
		tmp = t_0 * t_0
	else:
		tmp = math.cos((x * 2.0)) / ((c * (x * 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(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (x <= 4.5e-60)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(Float64(c * Float64(x * 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 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (x <= 4.5e-60)
		tmp = t_0 * t_0;
	else
		tmp = cos((x * 2.0)) / ((c * (x * 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[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.5e-60], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.50000000000000001e-60

   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* 65.4%

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

      2. unpow2 65.4%

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

      3. *-commutative 65.4%

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

      4. unpow2 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in x around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

      1. add-sqr-sqrt 59.9%

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

      2. sqrt-div 59.9%

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

      3. sqrt-div 59.9%

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

      4. metadata-eval 59.9%

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

      5. sqrt-prod 27.7%

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

      6. add-sqr-sqrt 39.1%

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

      7. associate-*r* 36.9%

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

      8. sqrt-prod 36.9%

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

      9. sqrt-unprod 10.9%

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

      10. add-sqr-sqrt 44.8%

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

      11. sqrt-prod 24.8%

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

      12. add-sqr-sqrt 45.2%

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

      13. *-commutative 45.2%

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

      14. sqrt-div 45.1%

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

      15. sqrt-div 45.7%

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

      16. metadata-eval 45.7%

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

      17. sqrt-prod 22.8%

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

      18. add-sqr-sqrt 41.9%

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

      19. associate-*r* 39.2%

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

      20. sqrt-prod 40.9%

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

   8. Applied egg-rr 83.8%

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

   if 4.50000000000000001e-60 < x

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*r* 55.7%

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

      3. associate-*r* 54.7%

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

      4. unpow2 54.7%

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

      5. unswap-sqr 65.5%

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

      6. unpow2 65.5%

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

      7. swap-sqr 97.8%

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

      8. *-commutative 97.8%

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

      9. *-commutative 97.8%

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

      10. *-commutative 97.8%

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

      11. *-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in s around 0 95.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;t_1 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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 (* s (* x c))) (t_1 (/ (/ 1.0 c) (* x s))))
   (if (<= x 1.15e-58) (* t_1 t_1) (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 1.15e-58) {
		tmp = t_1 * t_1;
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = (1.0d0 / c) / (x * s)
    if (x <= 1.15d-58) then
        tmp = t_1 * t_1
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = (1.0 / c) / (x * s);
	double tmp;
	if (x <= 1.15e-58) {
		tmp = t_1 * t_1;
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = (1.0 / c) / (x * s)
	tmp = 0
	if x <= 1.15e-58:
		tmp = t_1 * t_1
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = Float64(Float64(1.0 / c) / Float64(x * s))
	tmp = 0.0
	if (x <= 1.15e-58)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = (1.0 / c) / (x * s);
	tmp = 0.0;
	if (x <= 1.15e-58)
		tmp = t_1 * t_1;
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.15e-58], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999e-58

   1. Initial program 65.8%

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

      1. associate-/r* 65.6%

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

      2. unpow2 65.6%

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

      3. *-commutative 65.6%

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

      4. unpow2 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in x around 0 60.1%

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

      1. unpow2 60.1%

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

   6. Simplified 60.1%

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

      1. add-sqr-sqrt 60.1%

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

      2. sqrt-div 60.1%

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

      3. sqrt-div 60.1%

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

      4. metadata-eval 60.1%

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

      5. sqrt-prod 27.5%

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

      6. add-sqr-sqrt 38.9%

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

      7. associate-*r* 36.6%

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

      8. sqrt-prod 36.7%

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

      9. sqrt-unprod 10.9%

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

      10. add-sqr-sqrt 44.6%

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

      11. sqrt-prod 24.6%

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

      12. add-sqr-sqrt 44.9%

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

      13. *-commutative 44.9%

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

      14. sqrt-div 44.9%

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

      15. sqrt-div 45.5%

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

      16. metadata-eval 45.5%

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

      17. sqrt-prod 22.7%

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

      18. add-sqr-sqrt 42.2%

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

      19. associate-*r* 39.6%

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

      20. sqrt-prod 41.2%

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

   8. Applied egg-rr 83.9%

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

   if 1.1499999999999999e-58 < x

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*r* 55.2%

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

      3. associate-*r* 54.2%

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

      4. unpow2 54.2%

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

      5. unswap-sqr 65.1%

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

      6. unpow2 65.1%

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

      7. swap-sqr 97.8%

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

      8. *-commutative 97.8%

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

      9. *-commutative 97.8%

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

      10. *-commutative 97.8%

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

      11. *-commutative 97.8%

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

   3. Simplified 97.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 8: 80.0% 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 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.8%

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

   3. associate-*r* 58.4%

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

   4. unpow2 58.4%

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

   5. unswap-sqr 71.4%

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

   6. unpow2 71.4%

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

   7. swap-sqr 95.4%

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

   8. *-commutative 95.4%

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

   9. *-commutative 95.4%

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

   10. *-commutative 95.4%

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

   11. *-commutative 95.4%

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

3. Simplified 95.4%

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

4. Taylor expanded in x around 0 54.1%

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

   1. unpow2 54.1%

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

   2. associate-*r* 53.2%

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

   3. *-commutative 53.2%

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

   4. associate-*r* 54.0%

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

   5. unpow2 54.0%

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

   6. associate-/r* 54.2%

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

   7. unpow2 54.2%

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

   8. unpow2 54.2%

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

   9. unpow2 54.2%

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

6. Simplified 54.2%

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

   1. add-sqr-sqrt 54.2%

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

   2. pow2 54.2%

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

   3. sqrt-div 54.2%

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

   4. sqrt-div 54.2%

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

   5. metadata-eval 54.2%

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

   6. sqrt-prod 30.9%

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

   7. add-sqr-sqrt 58.0%

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

   8. unswap-sqr 69.3%

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

   9. *-commutative 69.3%

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

   10. *-commutative 69.3%

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

   11. sqrt-prod 45.9%

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

   12. add-sqr-sqrt 76.4%

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

   13. associate-/r* 76.3%

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

   14. pow2 76.3%

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

   15. inv-pow 76.3%

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

   16. inv-pow 76.3%

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

   17. pow-prod-up 76.4%

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

   18. metadata-eval 76.4%

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

   19. associate-*r* 77.4%

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

8. Applied egg-rr 77.3%

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

   1. metadata-eval 77.3%

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

   2. pow-prod-up 77.2%

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

   3. inv-pow 77.2%

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

   4. inv-pow 77.2%

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

   5. associate-*l* 76.3%

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

   6. associate-*l* 77.4%

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

10. Applied egg-rr 77.4%

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

11. Final simplification 77.4%

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

Alternative 9: 80.0% accurate, 20.9× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.4%

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

   1. associate-/r* 63.2%

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

   2. unpow2 63.2%

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

   3. *-commutative 63.2%

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

   4. unpow2 63.2%

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

3. Simplified 63.2%

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

4. Taylor expanded in x around 0 57.1%

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

   1. unpow2 57.1%

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

6. Simplified 57.1%

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

   1. add-sqr-sqrt 57.1%

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

   2. sqrt-div 57.1%

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

   3. sqrt-div 57.1%

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

   4. metadata-eval 57.1%

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

   5. sqrt-prod 27.2%

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

   6. add-sqr-sqrt 41.4%

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

   7. associate-*r* 39.4%

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

   8. sqrt-prod 39.5%

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

   9. sqrt-unprod 23.2%

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

   10. add-sqr-sqrt 45.1%

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

   11. sqrt-prod 24.5%

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

   12. add-sqr-sqrt 45.3%

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

   13. *-commutative 45.3%

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

   14. sqrt-div 45.2%

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

   15. sqrt-div 45.6%

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

   16. metadata-eval 45.6%

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

   17. sqrt-prod 24.3%

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

   18. add-sqr-sqrt 45.8%

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

   19. associate-*r* 43.6%

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

   20. sqrt-prod 44.9%

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

8. Applied egg-rr 77.4%

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

9. Final simplification 77.4%

   \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s} \]

Alternative 10: 55.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.8%

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

   3. associate-*r* 58.4%

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

   4. unpow2 58.4%

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

   5. unswap-sqr 71.4%

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

   6. unpow2 71.4%

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

   7. swap-sqr 95.4%

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

   8. *-commutative 95.4%

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

   9. *-commutative 95.4%

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

   10. *-commutative 95.4%

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

   11. *-commutative 95.4%

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

3. Simplified 95.4%

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

   1. associate-/r* 95.8%

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

   2. div-inv 95.8%

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

   3. *-commutative 95.8%

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

5. Applied egg-rr 95.8%

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

   1. un-div-inv 95.8%

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

   2. associate-*r* 93.5%

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

   3. *-commutative 93.5%

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

   4. associate-*r* 93.8%

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

   5. associate-*r* 95.0%

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

   6. *-commutative 95.0%

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

   7. associate-*r* 97.3%

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

7. Applied egg-rr 97.3%

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

8. Taylor expanded in x around 0 54.1%

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

   1. unpow2 54.1%

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

   2. unpow2 54.1%

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

   3. unpow2 54.1%

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

10. Simplified 54.1%

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

11. Final simplification 54.1%

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

Alternative 11: 78.7% accurate, 24.1× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.8%

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

   3. associate-*r* 58.4%

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

   4. unpow2 58.4%

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

   5. unswap-sqr 71.4%

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

   6. unpow2 71.4%

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

   7. swap-sqr 95.4%

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

   8. *-commutative 95.4%

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

   9. *-commutative 95.4%

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

   10. *-commutative 95.4%

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

   11. *-commutative 95.4%

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

3. Simplified 95.4%

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

   1. associate-/r* 95.8%

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

   2. div-inv 95.8%

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

   3. *-commutative 95.8%

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

5. Applied egg-rr 95.8%

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

   1. clear-num 95.8%

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

   2. frac-times 95.4%

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

   3. metadata-eval 95.4%

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

   4. associate-*r* 93.3%

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

   5. *-commutative 93.3%

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

   6. associate-*r* 93.3%

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

   7. associate-*r* 94.7%

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

   8. *-commutative 94.7%

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

   9. associate-*r* 96.9%

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

7. Applied egg-rr 96.9%

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

8. Taylor expanded in x around 0 76.1%

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

   1. *-commutative 76.1%

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

10. Simplified 76.1%

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

11. Final simplification 76.1%

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

Reproduce

herbie shell --seed 2023207 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))