mixedcos

Percentage Accurate: 64.8% → 98.5%

Time: 32.0s

Alternatives: 12

Speedup: 24.1×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -6.345523484422262e+49
  2. c = 4.1334891586105655e+249
  3. s = 6.759880953206116e+186

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: 64.8% 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: 98.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;s \leq 1.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{t_0}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t_0}{c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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 (* 2.0 x))))
   (if (<= s 1.6e+163)
     (/ t_0 (* (* c (* x s)) (* s (* x c))))
     (* (/ 1.0 x) (/ t_0 (* c (* s (* x (* c s)))))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (s <= 1.6e+163) {
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	} else {
		tmp = (1.0 / x) * (t_0 / (c * (s * (x * (c * s)))));
	}
	return tmp;
}

Fortran

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 = cos((2.0d0 * x))
    if (s <= 1.6d+163) then
        tmp = t_0 / ((c * (x * s)) * (s * (x * c)))
    else
        tmp = (1.0d0 / x) * (t_0 / (c * (s * (x * (c * s)))))
    end if
    code = tmp
end function

Java

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((2.0 * x));
	double tmp;
	if (s <= 1.6e+163) {
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	} else {
		tmp = (1.0 / x) * (t_0 / (c * (s * (x * (c * s)))));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if s <= 1.6e+163:
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)))
	else:
		tmp = (1.0 / x) * (t_0 / (c * (s * (x * (c * s)))))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (s <= 1.6e+163)
		tmp = Float64(t_0 / Float64(Float64(c * Float64(x * s)) * Float64(s * Float64(x * c))));
	else
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / Float64(c * Float64(s * Float64(x * Float64(c * s))))));
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (s <= 1.6e+163)
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	else
		tmp = (1.0 / x) * (t_0 / (c * (s * (x * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

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[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 1.6e+163], N[(t$95$0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / N[(c * N[(s * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.5999999999999999e163

   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%

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

      2. associate-*r* 63.0%

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

      3. associate-*r* 63.3%

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

      4. unpow2 63.3%

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

      5. unswap-sqr 78.5%

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

      6. unpow2 78.5%

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

      7. swap-sqr 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around 0 96.5%

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

   if 1.5999999999999999e163 < s

   1. Initial program 0.9%

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

      1. associate-*r* 0.9%

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

      2. *-commutative 0.9%

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

      3. associate-*r* 0.2%

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

      4. unpow2 0.2%

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

      5. unpow2 0.2%

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

   3. Simplified 0.2%

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

   4. Taylor expanded in c around 0 0.2%

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

      1. unpow2 0.2%

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

      2. *-commutative 0.2%

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

      3. associate-*r* 0.9%

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

      4. unpow2 0.9%

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

      5. associate-*l* 2.1%

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

      6. *-commutative 2.1%

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

      7. associate-*r* 61.9%

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

      8. associate-*l* 99.7%

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

      9. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.4%

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

      4. associate-*l* 99.5%

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

      5. *-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 2: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*l* 58.6%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. unpow2 59.3%

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

   6. associate-*r* 67.4%

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

   7. associate-*r* 68.7%

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

   8. *-commutative 68.7%

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

   9. unpow2 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in x around inf 58.8%

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

   1. unpow2 58.8%

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

   2. associate-/r* 58.6%

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

   3. *-commutative 58.6%

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

   4. unpow2 58.6%

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

   5. *-commutative 58.6%

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

   6. unpow2 58.6%

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

   7. swap-sqr 72.7%

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

   8. unpow2 72.7%

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

   9. associate-/r* 73.0%

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

   10. unpow2 73.0%

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

   11. swap-sqr 98.2%

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

   12. associate-/r* 98.2%

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

   13. *-rgt-identity 98.2%

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

   14. associate-*r/ 98.1%

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

6. Simplified 97.0%

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

7. Final simplification 97.0%

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

Alternative 3: 79.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;{\left(\frac{\frac{1 - x \cdot x}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.00072)
   (pow (/ (/ (- 1.0 (* x x)) c) (* x s)) 2.0)
   (if (<= x 8.2e+153)
     (/ (cos (* 2.0 x)) (* s (* (* c c) (* s (* x x)))))
     (/ 1.0 (pow (* c (* x s)) 2.0)))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.00072) {
		tmp = pow((((1.0 - (x * x)) / c) / (x * s)), 2.0);
	} else if (x <= 8.2e+153) {
		tmp = cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))));
	} else {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.00072d0) then
        tmp = (((1.0d0 - (x * x)) / c) / (x * s)) ** 2.0d0
    else if (x <= 8.2d+153) then
        tmp = cos((2.0d0 * x)) / (s * ((c * c) * (s * (x * x))))
    else
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    end if
    code = tmp
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.00072) {
		tmp = Math.pow((((1.0 - (x * x)) / c) / (x * s)), 2.0);
	} else if (x <= 8.2e+153) {
		tmp = Math.cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))));
	} else {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.00072:
		tmp = math.pow((((1.0 - (x * x)) / c) / (x * s)), 2.0)
	elif x <= 8.2e+153:
		tmp = math.cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))))
	else:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.00072)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / c) / Float64(x * s)) ^ 2.0;
	elseif (x <= 8.2e+153)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(s * Float64(Float64(c * c) * Float64(s * Float64(x * x)))));
	else
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.00072)
		tmp = (((1.0 - (x * x)) / c) / (x * s)) ^ 2.0;
	elseif (x <= 8.2e+153)
		tmp = cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))));
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.00072], N[Power[N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[x, 8.2e+153], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(c * c), $MachinePrecision] * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.20000000000000045e-4

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l* 62.0%

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

      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* 72.0%

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

      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%

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

   5. Applied egg-rr 87.9%

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

   6. Taylor expanded in x around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

   8. Simplified 81.6%

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

   if 7.20000000000000045e-4 < x < 8.20000000000000033e153

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 59.7%

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

      4. *-commutative 59.7%

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

      5. unpow2 59.7%

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

      6. associate-*r* 67.2%

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

      7. associate-*r* 81.2%

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

      8. *-commutative 81.2%

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

      9. unpow2 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in x around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. associate-*r* 82.0%

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

      4. unpow2 82.0%

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

      5. *-commutative 82.0%

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

   6. Simplified 82.0%

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

   if 8.20000000000000033e153 < x

   1. Initial program 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*l* 0.2%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. associate-*r* 0.0%

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

      7. associate-*r* 0.2%

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

      8. *-commutative 0.2%

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

      9. unpow2 0.2%

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

   3. Simplified 0.2%

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

   4. Taylor expanded in x around 0 0.2%

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

      1. unpow2 0.2%

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

      2. associate-/r* 0.2%

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

      3. unpow2 0.2%

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

      4. *-commutative 0.2%

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

      5. unpow2 0.2%

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

      6. swap-sqr 25.6%

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

      7. unpow2 25.6%

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

      8. associate-/r* 25.6%

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

      9. unpow2 25.6%

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

      10. swap-sqr 25.8%

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

      11. unpow2 25.8%

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

      12. associate-*r* 25.8%

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

      13. *-commutative 25.8%

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

   6. Simplified 25.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;{\left(\frac{\frac{1 - x \cdot x}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 4: 83.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;{\left(\frac{\frac{1 - x \cdot x}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.00072)
   (pow (/ (/ (- 1.0 (* x x)) c) (* x s)) 2.0)
   (/ (cos (* 2.0 x)) (* x (* c (* c (* s (* x s))))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.00072) {
		tmp = pow((((1.0 - (x * x)) / c) / (x * s)), 2.0);
	} else {
		tmp = cos((2.0 * x)) / (x * (c * (c * (s * (x * s)))));
	}
	return tmp;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.00072d0) then
        tmp = (((1.0d0 - (x * x)) / c) / (x * s)) ** 2.0d0
    else
        tmp = cos((2.0d0 * x)) / (x * (c * (c * (s * (x * s)))))
    end if
    code = tmp
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.00072) {
		tmp = Math.pow((((1.0 - (x * x)) / c) / (x * s)), 2.0);
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (c * (c * (s * (x * s)))));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.00072:
		tmp = math.pow((((1.0 - (x * x)) / c) / (x * s)), 2.0)
	else:
		tmp = math.cos((2.0 * x)) / (x * (c * (c * (s * (x * s)))))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.00072)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / c) / Float64(x * s)) ^ 2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(c * Float64(c * Float64(s * Float64(x * s))))));
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.00072)
		tmp = (((1.0 - (x * x)) / c) / (x * s)) ^ 2.0;
	else
		tmp = cos((2.0 * x)) / (x * (c * (c * (s * (x * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.00072], N[Power[N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000045e-4

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l* 62.0%

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

      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* 72.0%

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

      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%

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

   5. Applied egg-rr 87.9%

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

   6. Taylor expanded in x around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

   8. Simplified 81.6%

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

   if 7.20000000000000045e-4 < x

   1. Initial program 56.7%

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

      1. associate-*r* 55.2%

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

      2. *-commutative 55.2%

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

      3. associate-*r* 55.3%

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

      4. unpow2 55.3%

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

      5. unpow2 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in c around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

      3. associate-*r* 55.2%

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

      4. unpow2 55.2%

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

      5. associate-*l* 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-*r* 65.6%

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

      8. associate-*l* 72.1%

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

      9. *-commutative 72.1%

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

   6. Simplified 72.1%

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

   7. Taylor expanded in s around 0 65.6%

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

      1. unpow2 65.6%

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

      2. associate-*r* 65.6%

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

   9. Simplified 65.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00072:\\ \;\;\;\;{\left(\frac{\frac{1 - x \cdot x}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;s \leq 4.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{t_0}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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 (* 2.0 x))))
   (if (<= s 4.7e+181)
     (/ t_0 (* (* c (* x s)) (* s (* x c))))
     (/ t_0 (* x (* c (* (* x s) (* c s))))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (s <= 4.7e+181) {
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	} else {
		tmp = t_0 / (x * (c * ((x * s) * (c * s))));
	}
	return tmp;
}

Fortran

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 = cos((2.0d0 * x))
    if (s <= 4.7d+181) then
        tmp = t_0 / ((c * (x * s)) * (s * (x * c)))
    else
        tmp = t_0 / (x * (c * ((x * s) * (c * s))))
    end if
    code = tmp
end function

Java

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((2.0 * x));
	double tmp;
	if (s <= 4.7e+181) {
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	} else {
		tmp = t_0 / (x * (c * ((x * s) * (c * s))));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if s <= 4.7e+181:
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)))
	else:
		tmp = t_0 / (x * (c * ((x * s) * (c * s))))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (s <= 4.7e+181)
		tmp = Float64(t_0 / Float64(Float64(c * Float64(x * s)) * Float64(s * Float64(x * c))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(c * Float64(Float64(x * s) * Float64(c * s)))));
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (s <= 4.7e+181)
		tmp = t_0 / ((c * (x * s)) * (s * (x * c)));
	else
		tmp = t_0 / (x * (c * ((x * s) * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

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[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 4.7e+181], N[(t$95$0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 4.70000000000000027e181

   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. *-commutative 67.2%

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

      2. associate-*r* 62.0%

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

      3. associate-*r* 62.2%

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

      4. unpow2 62.2%

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

      5. unswap-sqr 77.2%

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

      6. unpow2 77.2%

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

      7. swap-sqr 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around 0 96.6%

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

   if 4.70000000000000027e181 < s

   1. Initial program 1.1%

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

      1. associate-*r* 1.1%

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

      2. *-commutative 1.1%

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

      3. associate-*r* 0.3%

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

      4. unpow2 0.3%

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

      5. unpow2 0.3%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in c around 0 0.3%

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

      1. unpow2 0.3%

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

      2. *-commutative 0.3%

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

      3. associate-*r* 1.1%

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

      4. unpow2 1.1%

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

      5. associate-*l* 2.2%

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

      6. *-commutative 2.2%

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

      7. associate-*r* 51.1%

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

      8. associate-*l* 99.6%

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

      9. *-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 6: 96.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq 1.95 \cdot 10^{+39}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (cos (* 2.0 x))))
   (if (<= x 1.95e+39)
     (/ t_1 (* x (* c (* (* x s) (* c s)))))
     (/ t_1 (* t_0 t_0)))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = cos((2.0 * x));
	double tmp;
	if (x <= 1.95e+39) {
		tmp = t_1 / (x * (c * ((x * s) * (c * s))));
	} else {
		tmp = t_1 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((2.0d0 * x))
    if (x <= 1.95d+39) then
        tmp = t_1 / (x * (c * ((x * s) * (c * s))))
    else
        tmp = t_1 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if (x <= 1.95e+39) {
		tmp = t_1 / (x * (c * ((x * s) * (c * s))));
	} else {
		tmp = t_1 / (t_0 * t_0);
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if x <= 1.95e+39:
		tmp = t_1 / (x * (c * ((x * s) * (c * s))))
	else:
		tmp = t_1 / (t_0 * t_0)
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (x <= 1.95e+39)
		tmp = Float64(t_1 / Float64(x * Float64(c * Float64(Float64(x * s) * Float64(c * s)))));
	else
		tmp = Float64(t_1 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if (x <= 1.95e+39)
		tmp = t_1 / (x * (c * ((x * s) * (c * s))));
	else
		tmp = t_1 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.95e+39], N[(t$95$1 / N[(x * N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.95e39

   1. Initial program 65.3%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

      3. associate-*r* 67.0%

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

      4. unpow2 67.0%

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

      5. unpow2 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in c around 0 67.0%

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

      1. unpow2 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-*r* 67.4%

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

      4. unpow2 67.4%

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

      5. associate-*l* 72.6%

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

      6. *-commutative 72.6%

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

      7. associate-*r* 83.7%

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

      8. associate-*l* 91.5%

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

      9. *-commutative 91.5%

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

   6. Simplified 91.5%

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

   if 1.95e39 < x

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

      2. associate-*r* 36.1%

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

      3. associate-*r* 36.3%

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

      4. unpow2 36.3%

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

      5. unswap-sqr 66.0%

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

      6. unpow2 66.0%

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

      7. swap-sqr 99.5%

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

      8. *-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. *-commutative 99.5%

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

      11. *-commutative 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+39}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 93.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. associate-*r* 65.0%

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

   2. *-commutative 65.0%

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

   3. associate-*r* 64.7%

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

   4. unpow2 64.7%

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

   5. unpow2 64.7%

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

3. Simplified 64.7%

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

4. Taylor expanded in c around 0 64.7%

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

   1. unpow2 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*r* 65.0%

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

   4. unpow2 65.0%

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

   5. associate-*l* 71.4%

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

   6. *-commutative 71.4%

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

   7. associate-*r* 80.9%

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

   8. associate-*l* 88.3%

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

   9. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Final simplification 88.3%

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

Alternative 8: 76.8% accurate, 2.9× speedup

Math

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

FPCore

NOTE: 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

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: 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

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

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

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

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: 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 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*l* 58.6%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. unpow2 59.3%

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

   6. associate-*r* 67.4%

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

   7. associate-*r* 68.7%

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

   8. *-commutative 68.7%

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

   9. unpow2 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in x around 0 49.2%

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

   1. unpow2 49.2%

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

   2. associate-/r* 49.0%

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

   3. unpow2 49.0%

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

   4. *-commutative 49.0%

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

   5. unpow2 49.0%

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

   6. swap-sqr 58.1%

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

   7. unpow2 58.1%

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

   8. associate-/r* 58.4%

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

   9. unpow2 58.4%

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

   10. swap-sqr 73.9%

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

   11. unpow2 73.9%

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

   12. associate-*r* 72.5%

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

   13. *-commutative 72.5%

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

6. Simplified 72.5%

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

7. Final simplification 72.5%

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

Alternative 9: 73.1% accurate, 20.9× speedup

Math

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

FPCore

NOTE: 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

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: 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

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

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

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

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: 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 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*l* 58.6%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. unpow2 59.3%

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

   6. associate-*r* 67.4%

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

   7. associate-*r* 68.7%

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

   8. *-commutative 68.7%

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

   9. unpow2 68.7%

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

3. Simplified 68.7%

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

   1. add-cube-cbrt 68.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

   2. times-frac 68.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

   3. associate-*r* 65.5%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

   4. swap-sqr 82.6%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

   5. associate-*r* 94.7%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

   6. *-commutative 94.7%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

   7. times-frac 94.7%

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

   8. associate-*l* 97.9%

      \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

   9. add-cube-cbrt 98.2%

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

   10. associate-/r* 98.2%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in x around 0 73.9%

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

7. Final simplification 73.9%

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

Alternative 10: 76.4% accurate, 20.9× speedup

Math

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

FPCore

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 (* x (* c s))))) (* t_0 t_0)))

C

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

Fortran

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

Java

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 / (x * (c * s));
	return t_0 * t_0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. associate-/r* 63.5%

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

   2. unpow2 63.5%

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

   3. *-commutative 63.5%

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

   4. unpow2 63.5%

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

3. Simplified 63.5%

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

4. Taylor expanded in x around 0 52.0%

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

   1. unpow2 52.0%

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

6. Simplified 52.0%

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

   1. associate-/r* 52.0%

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

   2. associate-*r* 49.3%

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

   3. add-sqr-sqrt 49.3%

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

8. Applied egg-rr 75.0%

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

9. Final simplification 75.0%

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

Alternative 11: 30.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ 0.6666666666666666 \cdot \left(\frac{x}{c \cdot c} \cdot \frac{\frac{x}{s}}{s}\right) \end{array} \]

FPCore

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
 (* 0.6666666666666666 (* (/ x (* c c)) (/ (/ x s) s))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 0.6666666666666666 * ((x / (c * c)) * ((x / s) / s));
}

Fortran

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 = 0.6666666666666666d0 * ((x / (c * c)) * ((x / s) / s))
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	return 0.6666666666666666 * ((x / (c * c)) * ((x / s) / s));
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 0.6666666666666666 * ((x / (c * c)) * ((x / s) / s))

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(0.6666666666666666 * Float64(Float64(x / Float64(c * c)) * Float64(Float64(x / s) / s)))
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 0.6666666666666666 * ((x / (c * c)) * ((x / s) / s));
end

Wolfram

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[(0.6666666666666666 * N[(N[(x / N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[(x / s), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*l* 58.6%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. unpow2 59.3%

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

   6. associate-*r* 67.4%

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

   7. associate-*r* 68.7%

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

   8. *-commutative 68.7%

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

   9. unpow2 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in x around 0 12.1%

   \[\leadsto \color{blue}{\left(\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} + 0.6666666666666666 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}\right) - 2 \cdot \frac{1}{{s}^{2} \cdot {c}^{2}}} \]
5. Step-by-step derivation

6. Simplified 12.1%

   \[\leadsto \color{blue}{\left(\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot s\right)\right)} + 0.6666666666666666 \cdot \frac{\frac{x \cdot x}{s \cdot s}}{c \cdot c}\right) - \frac{2}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}} \]

7. Taylor expanded in x around inf 23.5%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 23.5%

      \[\leadsto 0.6666666666666666 \cdot \frac{\color{blue}{x \cdot x}}{{c}^{2} \cdot {s}^{2}} \]

   2. associate-*r/ 23.5%

      \[\leadsto \color{blue}{\frac{0.6666666666666666 \cdot \left(x \cdot x\right)}{{c}^{2} \cdot {s}^{2}}} \]

   3. unpow2 23.5%

      \[\leadsto \frac{0.6666666666666666 \cdot \left(x \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

   4. unpow2 23.5%

      \[\leadsto \frac{0.6666666666666666 \cdot \left(x \cdot x\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

   5. associate-/r* 21.9%

      \[\leadsto \color{blue}{\frac{\frac{0.6666666666666666 \cdot \left(x \cdot x\right)}{c \cdot c}}{s \cdot s}} \]

   6. associate-*r/ 21.9%

      \[\leadsto \frac{\color{blue}{0.6666666666666666 \cdot \frac{x \cdot x}{c \cdot c}}}{s \cdot s} \]

   7. associate-*r/ 21.9%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{\frac{x \cdot x}{c \cdot c}}{s \cdot s}} \]

   8. associate-*l/ 29.4%

      \[\leadsto 0.6666666666666666 \cdot \frac{\color{blue}{\frac{x}{c \cdot c} \cdot x}}{s \cdot s} \]

   9. associate-*r/ 32.9%

      \[\leadsto 0.6666666666666666 \cdot \color{blue}{\left(\frac{x}{c \cdot c} \cdot \frac{x}{s \cdot s}\right)} \]

   10. associate-/r* 29.2%

       \[\leadsto 0.6666666666666666 \cdot \left(\frac{x}{c \cdot c} \cdot \color{blue}{\frac{\frac{x}{s}}{s}}\right) \]

9. Simplified 29.2%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot \left(\frac{x}{c \cdot c} \cdot \frac{\frac{x}{s}}{s}\right)} \]

10. Final simplification 29.2%

    \[\leadsto 0.6666666666666666 \cdot \left(\frac{x}{c \cdot c} \cdot \frac{\frac{x}{s}}{s}\right) \]

Alternative 12: 69.0% accurate, 24.1× speedup

Math

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

FPCore

NOTE: 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

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: 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

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

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

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

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: 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 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*l* 58.6%

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

   3. associate-*r* 59.3%

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

   4. *-commutative 59.3%

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

   5. unpow2 59.3%

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

   6. associate-*r* 67.4%

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

   7. associate-*r* 68.7%

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

   8. *-commutative 68.7%

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

   9. unpow2 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in x around 0 49.2%

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

   1. unpow2 49.2%

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

   2. unpow2 49.2%

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

   3. associate-*r* 50.3%

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

   4. *-commutative 50.3%

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

   5. associate-/r* 50.3%

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

   6. associate-/r* 50.3%

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

   7. associate-/r* 49.3%

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

   8. *-commutative 49.3%

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

   9. swap-sqr 61.5%

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

   10. unpow2 61.5%

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

   11. associate-/r* 61.5%

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

   12. unpow2 61.5%

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

   13. swap-sqr 49.3%

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

   14. *-commutative 49.3%

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

   15. unpow2 49.3%

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

   16. *-commutative 49.3%

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

6. Simplified 49.3%

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

7. Taylor expanded in x around 0 49.3%

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

   1. unpow2 49.3%

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

   2. unpow2 49.3%

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

   3. swap-sqr 61.5%

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

   4. unpow2 61.5%

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

9. Simplified 61.5%

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

    1. unpow2 61.5%

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

11. Applied egg-rr 61.5%

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

12. Final simplification 61.5%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))