mixedcos

Percentage Accurate: 66.3% → 99.2%

Time: 12.3s

Alternatives: 12

Speedup: 24.1×

• 32.0% 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: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 7.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{t_0}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* c (* x s))))
   (if (<= x 7.4e+44)
     (/ t_0 (* t_1 t_1))
     (* (/ 1.0 s) (/ t_0 (* (* x c) (* s (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 7.4e+44) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = c * (x * s)
    if (x <= 7.4d+44) then
        tmp = t_0 / (t_1 * t_1)
    else
        tmp = (1.0d0 / s) * (t_0 / ((x * c) * (s * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 7.4e+44) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = c * (x * s)
	tmp = 0
	if x <= 7.4e+44:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 7.4e+44)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(1.0 / s) * Float64(t_0 / Float64(Float64(x * c) * Float64(s * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 7.4e+44)
		tmp = t_0 / (t_1 * t_1);
	else
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.4e+44], N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / s), $MachinePrecision] * N[(t$95$0 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4000000000000002e44

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 58.2%

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

      3. associate-*r* 59.3%

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

      4. unpow2 59.3%

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

      5. unswap-sqr 75.0%

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

      6. unpow2 75.0%

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

      7. swap-sqr 98.0%

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

      8. *-commutative 98.0%

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

      9. *-commutative 98.0%

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

      10. *-commutative 98.0%

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

      11. *-commutative 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in s around 0 95.9%

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

   5. Taylor expanded in s around 0 97.2%

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

   if 7.4000000000000002e44 < x

   1. Initial program 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*r* 56.7%

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

      3. associate-*r* 58.5%

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

      4. unpow2 58.5%

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

      5. unswap-sqr 71.7%

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

      6. unpow2 71.7%

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

      7. swap-sqr 98.1%

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

      8. *-commutative 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

      11. *-commutative 98.1%

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

   3. Simplified 98.1%

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

      1. *-un-lft-identity 98.1%

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

      2. associate-*l* 97.9%

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

      3. times-frac 98.0%

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

      4. *-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 2: 93.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4.1e-5)
   (/ 1.0 (pow (* c (* x s)) 2.0))
   (/ (cos (* x 2.0)) (* s (* s (* x (* c (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.1e-5) {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	} else {
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4.1d-5) then
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (s * (s * (x * (c * (x * c)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.1e-5) {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4.1e-5:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	else:
		tmp = math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4.1e-5)
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(s * Float64(x * Float64(c * Float64(x * c))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4.1e-5)
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	else
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4.1e-5], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(x * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.10000000000000005e-5

   1. Initial program 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-*r* 56.6%

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

      3. associate-*r* 57.7%

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

      4. unpow2 57.7%

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

      5. unswap-sqr 74.2%

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

      6. unpow2 74.2%

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

      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. Taylor expanded in x around 0 53.2%

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

      1. unpow2 53.2%

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

      2. associate-*r* 54.7%

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

      3. *-commutative 54.7%

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

      4. associate-*r* 54.7%

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

      5. *-commutative 54.7%

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

      6. unpow2 54.7%

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

      7. unpow2 54.7%

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

      8. swap-sqr 69.6%

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

      9. swap-sqr 88.4%

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

      10. unpow2 88.4%

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

      11. associate-*r* 88.0%

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

      12. *-commutative 88.0%

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

      13. *-commutative 88.0%

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

      14. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   if 4.10000000000000005e-5 < x

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 63.2%

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

      4. *-commutative 63.2%

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

      5. unpow2 63.2%

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

      6. associate-*r* 69.5%

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

      7. associate-*r* 71.3%

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

      8. *-commutative 71.3%

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

      9. unpow2 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. associate-*r* 80.6%

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

      4. unpow2 80.6%

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

      5. associate-*r* 87.1%

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

      6. *-commutative 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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))))
   (if (<= x 2e-19)
     (/ 1.0 (pow (* c (* x s)) 2.0))
     (/ (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 tmp;
	if (x <= 2e-19) {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	} 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) :: tmp
    t_0 = s * (x * c)
    if (x <= 2d-19) then
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    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 tmp;
	if (x <= 2e-19) {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	} 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)
	tmp = 0
	if x <= 2e-19:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	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))
	tmp = 0.0
	if (x <= 2e-19)
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	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);
	tmp = 0.0;
	if (x <= 2e-19)
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	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]}, If[LessEqual[x, 2e-19], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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 < 2e-19

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*r* 56.4%

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

      3. associate-*r* 57.5%

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

      4. unpow2 57.5%

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

      5. unswap-sqr 74.1%

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

      6. unpow2 74.1%

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

      7. swap-sqr 97.9%

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

      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. Taylor expanded in x around 0 52.9%

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

      1. unpow2 52.9%

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

      2. associate-*r* 54.4%

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

      3. *-commutative 54.4%

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

      4. associate-*r* 54.5%

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

      5. *-commutative 54.5%

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

      6. unpow2 54.5%

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

      7. unpow2 54.5%

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

      8. swap-sqr 69.5%

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

      9. swap-sqr 88.3%

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

      10. unpow2 88.3%

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

      11. associate-*r* 87.9%

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

      12. *-commutative 87.9%

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

      13. *-commutative 87.9%

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

      14. *-commutative 87.9%

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

   6. Simplified 87.9%

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

   if 2e-19 < x

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*r* 62.4%

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

      3. associate-*r* 63.9%

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

      4. unpow2 63.9%

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

      5. unswap-sqr 74.9%

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

      6. unpow2 74.9%

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

      7. swap-sqr 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. *-commutative 98.4%

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

      11. *-commutative 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 4: 96.8% accurate, 2.7× speedup

Math

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

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 (* c (* x s)))) (/ (cos (* x 2.0)) (* t_0 t_0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. *-commutative 62.5%

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

   2. associate-*r* 57.9%

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

   3. associate-*r* 59.1%

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

   4. unpow2 59.1%

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

   5. unswap-sqr 74.3%

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

   6. unpow2 74.3%

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

   7. swap-sqr 98.0%

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

   8. *-commutative 98.0%

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

   9. *-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. *-commutative 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in s around 0 95.3%

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

5. Taylor expanded in s around 0 96.7%

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

6. Final simplification 96.7%

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

Alternative 5: 80.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. *-commutative 62.5%

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

   2. associate-*r* 57.9%

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

   3. associate-*r* 59.1%

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

   4. unpow2 59.1%

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

   5. unswap-sqr 74.3%

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

   6. unpow2 74.3%

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

   7. swap-sqr 98.0%

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

   8. *-commutative 98.0%

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

   9. *-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. *-commutative 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. associate-*r* 53.9%

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

   3. *-commutative 53.9%

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

   4. associate-*r* 54.0%

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

   5. *-commutative 54.0%

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

   6. unpow2 54.0%

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

   7. unpow2 54.0%

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

   8. swap-sqr 66.6%

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

   9. swap-sqr 82.7%

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

   10. unpow2 82.7%

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

   11. associate-*r* 82.2%

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

   12. *-commutative 82.2%

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

   13. *-commutative 82.2%

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

   14. *-commutative 82.2%

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

6. Simplified 82.2%

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

7. Final simplification 82.2%

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

Alternative 6: 80.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. *-commutative 62.5%

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

   2. associate-*r* 57.9%

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

   3. associate-*r* 59.1%

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

   4. unpow2 59.1%

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

   5. unswap-sqr 74.3%

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

   6. unpow2 74.3%

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

   7. swap-sqr 98.0%

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

   8. *-commutative 98.0%

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

   9. *-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. *-commutative 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. unpow2 52.8%

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

   3. *-commutative 52.8%

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

   4. unpow2 52.8%

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

6. Simplified 52.8%

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

   1. *-commutative 52.8%

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

   2. *-commutative 52.8%

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

   3. pow2 52.8%

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

   4. pow2 52.8%

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

   5. pow-prod-down 65.0%

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

   6. pow2 65.0%

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

   7. unpow-prod-down 82.2%

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

   8. associate-*r* 82.7%

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

   9. pow-flip 82.7%

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

   10. metadata-eval 82.7%

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

8. Applied egg-rr 82.7%

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

9. Taylor expanded in s around 0 82.2%

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

10. Final simplification 82.2%

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

Alternative 7: 78.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.38 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot s} \cdot \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 1.38e-191)
   (/ 1.0 (* (* c c) (* (* x s) (* x s))))
   (* (/ 1.0 (* c s)) (/ (/ 1.0 (* s (* x c))) x))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.38e-191) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = (1.0 / (c * s)) * ((1.0 / (s * (x * c))) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 1.38d-191) then
        tmp = 1.0d0 / ((c * c) * ((x * s) * (x * s)))
    else
        tmp = (1.0d0 / (c * s)) * ((1.0d0 / (s * (x * c))) / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.38e-191) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = (1.0 / (c * s)) * ((1.0 / (s * (x * c))) / x);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 1.38e-191:
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)))
	else:
		tmp = (1.0 / (c * s)) * ((1.0 / (s * (x * c))) / x)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 1.38e-191)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(x * s) * Float64(x * s))));
	else
		tmp = Float64(Float64(1.0 / Float64(c * s)) * Float64(Float64(1.0 / Float64(s * Float64(x * c))) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.38e-191)
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	else
		tmp = (1.0 / (c * s)) * ((1.0 / (s * (x * c))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 1.38e-191], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.38000000000000003e-191

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 57.4%

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

      3. associate-*r* 57.5%

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

      4. unpow2 57.5%

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

      5. unswap-sqr 74.2%

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

      6. unpow2 74.2%

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

      7. swap-sqr 97.4%

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

      8. *-commutative 97.4%

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

      9. *-commutative 97.4%

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

      10. *-commutative 97.4%

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

      11. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in s around 0 95.6%

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

      1. associate-*r* 97.5%

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

      2. *-commutative 97.5%

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

      3. swap-sqr 74.8%

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

      4. *-commutative 74.8%

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

      5. *-commutative 74.8%

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

      6. swap-sqr 57.4%

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

      7. associate-*r* 62.0%

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

      8. /-rgt-identity 62.0%

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

      9. clear-num 61.4%

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

      10. div-inv 61.4%

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

      11. clear-num 61.4%

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

      12. div-inv 61.1%

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

      13. frac-times 62.0%

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

      14. associate-*r* 57.4%

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

      15. swap-sqr 74.8%

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

      16. *-commutative 74.8%

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

      17. *-commutative 74.8%

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

      18. swap-sqr 97.5%

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

      19. *-commutative 97.5%

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

      20. associate-*r* 95.6%

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

      21. *-commutative 95.6%

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

      22. associate-*r* 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in x around 0 52.8%

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

      1. unpow2 52.8%

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

      2. unpow2 52.8%

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

      3. associate-*l* 59.3%

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

      4. associate-*l* 60.5%

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

      5. *-commutative 60.5%

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

      6. unpow2 60.5%

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

      7. associate-*r* 60.7%

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

      8. unpow2 60.7%

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

      9. *-commutative 60.7%

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

      10. associate-*r* 54.1%

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

      11. unpow2 54.1%

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

      12. associate-*r* 52.8%

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

      13. unpow2 52.8%

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

      14. unpow2 52.8%

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

      15. swap-sqr 66.9%

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

   9. Simplified 66.9%

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

   if 1.38000000000000003e-191 < x

   1. Initial program 63.1%

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

      1. associate-/r* 61.4%

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

      2. unpow2 61.4%

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

      3. *-commutative 61.4%

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

      4. unpow2 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in x around 0 54.3%

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

      1. unpow2 54.3%

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

   6. Simplified 54.3%

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

      1. div-inv 54.3%

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

      2. frac-times 55.1%

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

      3. associate-*r* 52.8%

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

      4. swap-sqr 62.7%

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

      5. *-commutative 62.7%

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

      6. *-commutative 62.7%

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

      7. swap-sqr 78.4%

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

      8. *-commutative 78.4%

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

      9. associate-*r* 78.5%

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

      10. *-commutative 78.5%

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

      11. associate-*r* 80.4%

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

      12. frac-times 80.3%

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

      13. associate-*l/ 80.3%

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

      14. associate-*r* 78.5%

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

      15. *-commutative 78.5%

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

      16. associate-*r* 80.3%

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

      17. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.38 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot s} \cdot \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x}\\ \end{array} \]

Alternative 8: 74.3% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 115000:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 115000.0)
   (/ 1.0 (* (* c c) (* (* x s) (* x s))))
   (/ (/ 1.0 (* s (* x s))) (* c (* x c)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 115000.0) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = (1.0 / (s * (x * s))) / (c * (x * c));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 115000.0d0) then
        tmp = 1.0d0 / ((c * c) * ((x * s) * (x * s)))
    else
        tmp = (1.0d0 / (s * (x * s))) / (c * (x * c))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 115000.0) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = (1.0 / (s * (x * s))) / (c * (x * c));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 115000.0:
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)))
	else:
		tmp = (1.0 / (s * (x * s))) / (c * (x * c))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 115000.0)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(x * s) * Float64(x * s))));
	else
		tmp = Float64(Float64(1.0 / Float64(s * Float64(x * s))) / Float64(c * Float64(x * c)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 115000.0)
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	else
		tmp = (1.0 / (s * (x * s))) / (c * (x * c));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 115000.0], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 115000

   1. Initial program 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-*r* 57.0%

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

      3. associate-*r* 58.1%

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

      4. unpow2 58.1%

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

      5. unswap-sqr 74.5%

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

      6. unpow2 74.5%

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

      7. swap-sqr 97.9%

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

      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. Taylor expanded in s around 0 95.7%

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

      1. associate-*r* 97.1%

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

      2. *-commutative 97.1%

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

      3. swap-sqr 72.4%

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

      4. *-commutative 72.4%

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

      5. *-commutative 72.4%

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

      6. swap-sqr 57.0%

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

      7. associate-*r* 60.9%

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

      8. /-rgt-identity 60.9%

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

      9. clear-num 60.5%

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

      10. div-inv 60.5%

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

      11. clear-num 60.5%

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

      12. div-inv 60.3%

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

      13. frac-times 60.9%

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

      14. associate-*r* 57.0%

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

      15. swap-sqr 72.4%

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

      16. *-commutative 72.4%

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

      17. *-commutative 72.4%

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

      18. swap-sqr 97.1%

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

      19. *-commutative 97.1%

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

      20. associate-*r* 95.7%

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

      21. *-commutative 95.7%

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

      22. associate-*r* 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x around 0 53.7%

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

      1. unpow2 53.7%

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

      2. unpow2 53.7%

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

      3. associate-*l* 60.5%

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

      4. associate-*l* 62.8%

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

      5. *-commutative 62.8%

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

      6. unpow2 62.8%

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

      7. associate-*r* 61.1%

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

      8. unpow2 61.1%

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

      9. *-commutative 61.1%

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

      10. associate-*r* 55.2%

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

      11. unpow2 55.2%

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

      12. associate-*r* 53.7%

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

      13. unpow2 53.7%

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

      14. unpow2 53.7%

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

      15. swap-sqr 66.6%

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

   9. Simplified 66.6%

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

   if 115000 < x

   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. unpow2 67.2%

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

      2. *-commutative 67.2%

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

      3. unpow2 67.2%

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

   3. Simplified 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. associate-*r* 68.8%

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

      3. times-frac 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

   5. Applied egg-rr 68.8%

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

      1. associate-*l/ 68.8%

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

      2. *-lft-identity 68.8%

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

      3. associate-/r* 68.8%

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

      4. unpow2 68.8%

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

      5. *-commutative 68.8%

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

      6. unpow2 68.8%

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

      7. associate-*l* 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in x around 0 56.2%

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

      1. unpow2 56.2%

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

      2. associate-*l* 61.9%

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

   10. Simplified 61.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 115000:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 9: 74.3% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s \cdot s}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4e+56)
   (/ 1.0 (* (* c c) (* (* x s) (* x s))))
   (/ (/ (/ 1.0 x) (* s s)) (* c (* x c)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e+56) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = ((1.0 / x) / (s * s)) / (c * (x * c));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4d+56) then
        tmp = 1.0d0 / ((c * c) * ((x * s) * (x * s)))
    else
        tmp = ((1.0d0 / x) / (s * s)) / (c * (x * c))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e+56) {
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	} else {
		tmp = ((1.0 / x) / (s * s)) / (c * (x * c));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4e+56:
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)))
	else:
		tmp = ((1.0 / x) / (s * s)) / (c * (x * c))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4e+56)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(x * s) * Float64(x * s))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(s * s)) / Float64(c * Float64(x * c)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4e+56)
		tmp = 1.0 / ((c * c) * ((x * s) * (x * s)));
	else
		tmp = ((1.0 / x) / (s * s)) / (c * (x * c));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4e+56], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(s * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000037e56

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*r* 58.4%

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

      3. associate-*r* 59.5%

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

      4. unpow2 59.5%

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

      5. unswap-sqr 75.1%

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

      6. unpow2 75.1%

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

      7. swap-sqr 98.0%

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

      8. *-commutative 98.0%

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

      9. *-commutative 98.0%

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

      10. *-commutative 98.0%

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

      11. *-commutative 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in s around 0 95.9%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. swap-sqr 73.2%

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

      4. *-commutative 73.2%

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

      5. *-commutative 73.2%

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

      6. swap-sqr 58.4%

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

      7. associate-*r* 62.2%

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

      8. /-rgt-identity 62.2%

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

      9. clear-num 61.8%

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

      10. div-inv 61.8%

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

      11. clear-num 61.8%

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

      12. div-inv 61.6%

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

      13. frac-times 62.2%

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

      14. associate-*r* 58.4%

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

      15. swap-sqr 73.2%

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

      16. *-commutative 73.2%

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

      17. *-commutative 73.2%

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

      18. swap-sqr 97.2%

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

      19. *-commutative 97.2%

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

      20. associate-*r* 95.9%

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

      21. *-commutative 95.9%

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

      22. associate-*r* 98.0%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around 0 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*l* 60.4%

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

      4. associate-*l* 62.7%

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

      5. *-commutative 62.7%

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

      6. unpow2 62.7%

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

      7. associate-*r* 61.0%

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

      8. unpow2 61.0%

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

      9. *-commutative 61.0%

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

      10. associate-*r* 55.3%

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

      11. unpow2 55.3%

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

      12. associate-*r* 53.9%

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

      13. unpow2 53.9%

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

      14. unpow2 53.9%

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

      15. swap-sqr 66.2%

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

   9. Simplified 66.2%

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

   if 4.00000000000000037e56 < x

   1. Initial program 63.7%

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

      1. unpow2 63.7%

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

      2. *-commutative 63.7%

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

      3. unpow2 63.7%

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

   3. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. associate-*r* 65.5%

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

      3. times-frac 65.5%

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

      4. *-commutative 65.5%

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

      5. *-commutative 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.5%

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

      2. *-lft-identity 65.5%

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

      3. associate-/r* 65.5%

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

      4. unpow2 65.5%

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

      5. *-commutative 65.5%

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

      6. unpow2 65.5%

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

      7. associate-*l* 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in x around 0 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-/r* 55.8%

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

      3. unpow2 55.8%

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

   10. Simplified 55.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s \cdot s}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 10: 78.2% accurate, 20.9× speedup

Math

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

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

Derivation

1. Initial program 62.5%

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

   1. associate-/r* 61.4%

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

   2. unpow2 61.4%

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

   3. *-commutative 61.4%

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

   4. unpow2 61.4%

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

3. Simplified 61.4%

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

4. Taylor expanded in x around 0 55.5%

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

   1. unpow2 55.5%

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

6. Simplified 55.5%

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

   1. div-inv 55.3%

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

   2. frac-times 56.2%

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

   3. associate-*r* 52.8%

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

   4. swap-sqr 65.0%

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

   5. *-commutative 65.0%

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

   6. *-commutative 65.0%

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

   7. swap-sqr 82.2%

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

   8. *-commutative 82.2%

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

   9. associate-*r* 81.5%

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

   10. *-commutative 81.5%

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

   11. associate-*r* 82.7%

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

   12. frac-times 82.6%

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

8. Applied egg-rr 82.6%

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

9. Final simplification 82.6%

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

Alternative 11: 70.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. *-commutative 62.5%

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

   2. associate-*r* 57.9%

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

   3. associate-*r* 59.1%

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

   4. unpow2 59.1%

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

   5. unswap-sqr 74.3%

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

   6. unpow2 74.3%

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

   7. swap-sqr 98.0%

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

   8. *-commutative 98.0%

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

   9. *-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. *-commutative 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in s around 0 95.3%

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

   1. associate-*r* 96.7%

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

   2. *-commutative 96.7%

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

   3. swap-sqr 75.3%

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

   4. *-commutative 75.3%

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

   5. *-commutative 75.3%

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

   6. swap-sqr 57.9%

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

   7. associate-*r* 62.5%

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

   8. /-rgt-identity 62.5%

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

   9. clear-num 61.4%

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

   10. div-inv 61.4%

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

   11. clear-num 61.4%

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

   12. div-inv 61.3%

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

   13. frac-times 62.5%

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

   14. associate-*r* 57.9%

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

   15. swap-sqr 75.3%

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

   16. *-commutative 75.3%

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

   17. *-commutative 75.3%

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

   18. swap-sqr 96.7%

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

   19. *-commutative 96.7%

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

   20. associate-*r* 95.3%

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

   21. *-commutative 95.3%

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

   22. associate-*r* 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. unpow2 52.8%

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

   3. associate-*l* 59.6%

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

   4. associate-*l* 61.4%

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

   5. *-commutative 61.4%

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

   6. unpow2 61.4%

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

   7. associate-*r* 60.0%

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

   8. unpow2 60.0%

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

   9. *-commutative 60.0%

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

   10. associate-*r* 53.9%

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

   11. unpow2 53.9%

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

   12. associate-*r* 52.8%

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

   13. unpow2 52.8%

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

   14. unpow2 52.8%

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

   15. swap-sqr 65.0%

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

9. Simplified 65.0%

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

10. Final simplification 65.0%

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

Alternative 12: 25.9% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. *-commutative 62.5%

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

   2. associate-*r* 57.9%

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

   3. associate-*r* 59.1%

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

   4. unpow2 59.1%

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

   5. unswap-sqr 74.3%

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

   6. unpow2 74.3%

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

   7. swap-sqr 98.0%

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

   8. *-commutative 98.0%

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

   9. *-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. *-commutative 98.0%

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

3. Simplified 98.0%

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

   1. *-un-lft-identity 98.0%

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

   2. associate-*l* 94.8%

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

   3. times-frac 94.8%

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

   4. *-commutative 94.8%

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

5. Applied egg-rr 94.8%

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

6. Taylor expanded in x around 0 40.6%

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

   1. associate-*r* 40.1%

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

   2. *-commutative 40.1%

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

   3. associate-*r* 40.5%

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

   4. associate-/r* 40.4%

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

   5. unpow2 40.4%

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

   6. unpow2 40.4%

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

   7. associate-*r/ 40.4%

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

   8. metadata-eval 40.4%

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

   9. *-commutative 40.4%

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

   10. unpow2 40.4%

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

8. Simplified 40.4%

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

9. Taylor expanded in x around inf 27.8%

   \[\leadsto \frac{1}{s} \cdot \color{blue}{\frac{-2}{{c}^{2} \cdot s}} \]
10. Step-by-step derivation

    1. unpow2 27.8%

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

    2. associate-*l* 25.5%

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

11. Simplified 25.5%

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

12. Final simplification 25.5%

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

Reproduce

herbie shell --seed 2023199 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))