mixedcos

Percentage Accurate: 67.3% → 96.9%

Time: 10.3s

Alternatives: 10

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c)))) (/ (cos (* 2.0 x)) (* t_0 t_0))))

C

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

Fortran

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 = s * (x * c)
    code = cos((2.0d0 * x)) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = s * (x * c);
	tmp = cos((2.0 * x)) / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*r* 62.4%

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

   3. associate-*r* 61.7%

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

   4. unpow2 61.7%

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

   5. unswap-sqr 79.8%

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

   6. unpow2 79.8%

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

   7. swap-sqr 98.8%

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

   8. *-commutative 98.8%

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

   9. *-commutative 98.8%

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

   10. *-commutative 98.8%

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

   11. *-commutative 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{c \cdot \left(x \cdot s\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (pow s 2.0) 5e+153)
   (/ (cos (* 2.0 x)) (* s (* (* c c) (* s (* x x)))))
   (pow (/ 1.0 (* c (* x s))) 2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (pow(s, 2.0) <= 5e+153) {
		tmp = cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))));
	} else {
		tmp = pow((1.0 / (c * (x * s))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((s ** 2.0d0) <= 5d+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

public static double code(double x, double c, double s) {
	double tmp;
	if (Math.pow(s, 2.0) <= 5e+153) {
		tmp = Math.cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))));
	} else {
		tmp = Math.pow((1.0 / (c * (x * s))), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if math.pow(s, 2.0) <= 5e+153:
		tmp = math.cos((2.0 * x)) / (s * ((c * c) * (s * (x * x))))
	else:
		tmp = math.pow((1.0 / (c * (x * s))), 2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if ((s ^ 2.0) <= 5e+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

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((s ^ 2.0) <= 5e+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

code[x_, c_, s_] := If[LessEqual[N[Power[s, 2.0], $MachinePrecision], 5e+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[Power[N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 5.00000000000000018e153

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*l* 67.7%

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

      3. associate-*r* 66.6%

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

      4. *-commutative 66.6%

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

      5. unpow2 66.6%

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

      6. associate-*r* 71.5%

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

      7. associate-*r* 72.7%

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

      8. *-commutative 72.7%

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

      9. unpow2 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around 0 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. associate-*r* 73.8%

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

      4. unpow2 73.8%

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

      5. *-commutative 73.8%

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

   6. Simplified 73.8%

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

   if 5.00000000000000018e153 < (pow.f64 s 2)

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 52.0%

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

      3. associate-*r* 51.1%

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

      4. *-commutative 51.1%

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

      5. unpow2 51.1%

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

      6. associate-*r* 59.4%

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

      7. associate-*r* 59.9%

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

      8. *-commutative 59.9%

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

      9. unpow2 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in x around 0 51.0%

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

      1. associate-/r* 51.0%

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

      2. unpow2 51.0%

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

      3. *-commutative 51.0%

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

      4. unpow2 51.0%

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

      5. swap-sqr 72.7%

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

      6. unpow2 72.7%

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

      7. associate-/l/ 72.7%

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

      8. unpow2 72.7%

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

      9. swap-sqr 51.0%

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

      10. unpow2 51.0%

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

      11. *-commutative 51.0%

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

      12. associate-*r* 52.0%

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

      13. unpow2 52.0%

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

      14. *-commutative 52.0%

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

      15. unpow2 52.0%

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

   6. Simplified 52.0%

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

      1. inv-pow 52.0%

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

      2. associate-*r* 51.1%

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

      3. pow2 51.1%

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

      4. pow2 51.1%

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

      5. unpow-prod-down 70.6%

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

      6. pow2 70.6%

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

      7. pow-prod-down 90.0%

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

      8. associate-*r* 91.9%

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

      9. pow2 91.9%

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

      10. pow-prod-down 91.9%

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

      11. metadata-eval 91.9%

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

      12. metadata-eval 91.9%

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

      13. pow2 91.9%

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

      14. metadata-eval 91.9%

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

      15. unpow-1 91.9%

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

   8. Applied egg-rr 91.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{c \cdot \left(x \cdot s\right)}\right)}^{2}\\ \end{array} \]

Alternative 3: 72.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.08 \cdot 10^{+123}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.08e+123)
   (/ (cos (* 2.0 x)) (* x (* x (* (* s s) (* c c)))))
   (pow (* c (* x s)) -2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.08e+123) {
		tmp = cos((2.0 * x)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 1.08d+123) then
        tmp = cos((2.0d0 * x)) / (x * (x * ((s * s) * (c * c))))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.08e+123) {
		tmp = Math.cos((2.0 * x)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.08e+123:
		tmp = math.cos((2.0 * x)) / (x * (x * ((s * s) * (c * c))))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.08e+123)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(x * Float64(Float64(s * s) * Float64(c * c)))));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 1.08e+123)
		tmp = cos((2.0 * x)) / (x * (x * ((s * s) * (c * c))));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.08e+123], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.0799999999999999e123

   1. Initial program 71.0%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-*r* 69.8%

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

      5. *-commutative 69.8%

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

      6. unpow2 69.8%

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

      7. unpow2 69.8%

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

   3. Simplified 69.8%

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

   if 1.0799999999999999e123 < s

   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* 54.9%

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

      3. associate-*r* 54.9%

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

      4. *-commutative 54.9%

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

      5. unpow2 54.9%

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

      6. associate-*r* 59.0%

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

      7. associate-*r* 58.6%

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

      8. *-commutative 58.6%

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

      9. unpow2 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x around 0 54.8%

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

      1. associate-/r* 54.8%

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

      2. unpow2 54.8%

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

      3. *-commutative 54.8%

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

      4. unpow2 54.8%

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

      5. swap-sqr 82.3%

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

      6. unpow2 82.3%

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

      7. associate-/l/ 82.5%

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

      8. unpow2 82.5%

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

      9. swap-sqr 54.8%

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

      10. unpow2 54.8%

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

      11. *-commutative 54.8%

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

      12. associate-*r* 54.9%

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

      13. unpow2 54.9%

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

      14. *-commutative 54.9%

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

      15. unpow2 54.9%

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

   6. Simplified 54.9%

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

      1. inv-pow 54.9%

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

      2. associate-*r* 54.9%

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

      3. pow2 54.9%

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

      4. pow2 54.9%

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

      5. unpow-prod-down 76.2%

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

      6. unpow-prod-down 72.9%

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

      7. inv-pow 72.9%

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

      8. pow-flip 72.9%

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

      9. metadata-eval 72.9%

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

      10. pow2 72.9%

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

      11. pow-pow 72.9%

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

      12. metadata-eval 72.9%

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

      13. unpow-prod-down 95.3%

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

      14. associate-*r* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.08 \cdot 10^{+123}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{c \cdot \left(x \cdot s\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.9e+144)
   (/ (cos (* 2.0 x)) (* x (* (* c (* x c)) (* s s))))
   (pow (/ 1.0 (* c (* x s))) 2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.9e+144) {
		tmp = cos((2.0 * x)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = pow((1.0 / (c * (x * s))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 1.9d+144) then
        tmp = cos((2.0d0 * x)) / (x * ((c * (x * c)) * (s * s)))
    else
        tmp = (1.0d0 / (c * (x * s))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.9e+144) {
		tmp = Math.cos((2.0 * x)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = Math.pow((1.0 / (c * (x * s))), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.9e+144:
		tmp = math.cos((2.0 * x)) / (x * ((c * (x * c)) * (s * s)))
	else:
		tmp = math.pow((1.0 / (c * (x * s))), 2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.9e+144)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(x * s))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 1.9e+144)
		tmp = cos((2.0 * x)) / (x * ((c * (x * c)) * (s * s)));
	else
		tmp = (1.0 / (c * (x * s))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.9e+144], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.90000000000000013e144

   1. Initial program 70.9%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*r* 70.3%

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

      4. unpow2 70.3%

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

      5. unpow2 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in c around 0 70.3%

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

      1. unpow2 70.3%

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

      2. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   if 1.90000000000000013e144 < s

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. unpow2 53.7%

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

      6. associate-*r* 58.2%

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

      7. associate-*r* 57.8%

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

      8. *-commutative 57.8%

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

      9. unpow2 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 53.6%

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

      1. associate-/r* 53.6%

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

      2. unpow2 53.6%

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

      3. *-commutative 53.6%

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

      4. unpow2 53.6%

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

      5. swap-sqr 82.1%

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

      6. unpow2 82.1%

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

      7. associate-/l/ 82.1%

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

      8. unpow2 82.1%

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

      9. swap-sqr 53.6%

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

      10. unpow2 53.6%

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

      11. *-commutative 53.6%

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

      12. associate-*r* 53.7%

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

      13. unpow2 53.7%

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

      14. *-commutative 53.7%

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

      15. unpow2 53.7%

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

   6. Simplified 53.7%

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

      1. inv-pow 53.7%

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

      2. associate-*r* 53.7%

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

      3. pow2 53.7%

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

      4. pow2 53.7%

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

      5. unpow-prod-down 75.1%

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

      6. pow2 75.1%

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

      7. pow-prod-down 96.5%

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

      8. associate-*r* 99.8%

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

      9. pow2 99.8%

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

      10. pow-prod-down 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. pow2 99.9%

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

      14. metadata-eval 99.9%

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

      15. unpow-1 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{c \cdot \left(x \cdot s\right)}\right)}^{2}\\ \end{array} \]

Alternative 5: 94.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*r* 62.4%

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

   3. associate-*r* 61.7%

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

   4. unpow2 61.7%

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

   5. unswap-sqr 79.8%

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

   6. unpow2 79.8%

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

   7. swap-sqr 98.8%

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

   8. *-commutative 98.8%

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

   9. *-commutative 98.8%

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

   10. *-commutative 98.8%

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

   11. *-commutative 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 96.6%

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

5. Final simplification 96.6%

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

Alternative 6: 78.5% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (pow (/ 1.0 (* c (* x s))) 2.0))

C

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

Fortran

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

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

Python

def code(x, c, s):
	return math.pow((1.0 / (c * (x * s))), 2.0)

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(c * Float64(x * s))) ^ 2.0
end

MATLAB

function tmp = code(x, c, s)
	tmp = (1.0 / (c * (x * s))) ^ 2.0;
end

Wolfram

code[x_, c_, s_] := N[Power[N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*l* 62.4%

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

   3. associate-*r* 61.3%

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

   4. *-commutative 61.3%

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

   5. unpow2 61.3%

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

   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* 68.3%

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

   8. *-commutative 68.3%

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

   9. unpow2 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. unpow2 55.5%

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

   3. *-commutative 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 69.0%

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

   6. unpow2 69.0%

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

   7. associate-/l/ 69.0%

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

   8. unpow2 69.0%

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

   9. swap-sqr 55.6%

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

   10. unpow2 55.6%

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

   11. *-commutative 55.6%

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

   12. associate-*r* 55.5%

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

   13. unpow2 55.5%

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

   14. *-commutative 55.5%

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

   15. unpow2 55.5%

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

6. Simplified 55.5%

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

   1. inv-pow 55.5%

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

   2. associate-*r* 55.6%

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

   3. pow2 55.6%

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

   4. pow2 55.6%

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

   5. unpow-prod-down 67.0%

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

   6. pow2 67.0%

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

   7. pow-prod-down 79.6%

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

   8. associate-*r* 79.5%

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

   9. pow2 79.5%

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

   10. pow-prod-down 79.7%

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

   11. metadata-eval 79.7%

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

   12. metadata-eval 79.7%

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

   13. pow2 79.7%

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

   14. metadata-eval 79.7%

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

   15. unpow-1 79.7%

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

8. Applied egg-rr 79.7%

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

9. Final simplification 79.7%

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

Alternative 7: 78.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*l* 62.4%

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

   3. associate-*r* 61.3%

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

   4. *-commutative 61.3%

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

   5. unpow2 61.3%

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

   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* 68.3%

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

   8. *-commutative 68.3%

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

   9. unpow2 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. unpow2 55.5%

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

   3. *-commutative 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 69.0%

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

   6. unpow2 69.0%

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

   7. associate-/l/ 69.0%

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

   8. unpow2 69.0%

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

   9. swap-sqr 55.6%

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

   10. unpow2 55.6%

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

   11. *-commutative 55.6%

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

   12. associate-*r* 55.5%

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

   13. unpow2 55.5%

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

   14. *-commutative 55.5%

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

   15. unpow2 55.5%

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

6. Simplified 55.5%

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

   1. inv-pow 55.5%

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

   2. associate-*r* 55.6%

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

   3. pow2 55.6%

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

   4. pow2 55.6%

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

   5. unpow-prod-down 67.0%

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

   6. unpow-prod-down 66.5%

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

   7. inv-pow 66.5%

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

   8. pow-flip 66.5%

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

   9. metadata-eval 66.5%

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

   10. pow2 66.5%

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

   11. pow-pow 66.5%

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

   12. metadata-eval 66.5%

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

   13. unpow-prod-down 79.6%

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

   14. associate-*r* 79.7%

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

8. Applied egg-rr 79.7%

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

9. Final simplification 79.7%

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

Alternative 8: 55.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c c) (* (* s s) (* x x)))))

C

double code(double x, double c, double s) {
	return 1.0 / ((c * c) * ((s * s) * (x * 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 = 1.0d0 / ((c * c) * ((s * s) * (x * x)))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((c * c) * ((s * s) * (x * x)));
}

Python

def code(x, c, s):
	return 1.0 / ((c * c) * ((s * s) * (x * x)))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(s * s) * Float64(x * x))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*l* 62.4%

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

   3. associate-*r* 61.3%

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

   4. *-commutative 61.3%

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

   5. unpow2 61.3%

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

   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* 68.3%

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

   8. *-commutative 68.3%

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

   9. unpow2 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. unpow2 55.5%

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

   3. *-commutative 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 69.0%

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

   6. unpow2 69.0%

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

   7. associate-/l/ 69.0%

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

   8. unpow2 69.0%

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

   9. swap-sqr 55.6%

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

   10. unpow2 55.6%

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

   11. *-commutative 55.6%

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

   12. associate-*r* 55.5%

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

   13. unpow2 55.5%

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

   14. *-commutative 55.5%

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

   15. unpow2 55.5%

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

6. Simplified 55.5%

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

7. Final simplification 55.5%

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

Alternative 9: 77.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c (* x s)) (* x (* s c)))))

C

double code(double x, double c, double s) {
	return 1.0 / ((c * (x * s)) * (x * (s * c)));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * (x * s)) * (x * (s * c)))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((c * (x * s)) * (x * (s * c)));
}

Python

def code(x, c, s):
	return 1.0 / ((c * (x * s)) * (x * (s * c)))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * Float64(x * s)) * Float64(x * Float64(s * c))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((c * (x * s)) * (x * (s * c)));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*l* 62.4%

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

   3. associate-*r* 61.3%

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

   4. *-commutative 61.3%

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

   5. unpow2 61.3%

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

   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* 68.3%

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

   8. *-commutative 68.3%

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

   9. unpow2 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. unpow2 55.5%

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

   3. *-commutative 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 69.0%

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

   6. unpow2 69.0%

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

   7. associate-/l/ 69.0%

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

   8. unpow2 69.0%

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

   9. swap-sqr 55.6%

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

   10. unpow2 55.6%

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

   11. *-commutative 55.6%

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

   12. associate-*r* 55.5%

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

   13. unpow2 55.5%

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

   14. *-commutative 55.5%

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

   15. unpow2 55.5%

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

6. Simplified 55.5%

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

7. Taylor expanded in c around 0 55.6%

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

   1. *-commutative 55.6%

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

   2. unpow2 55.6%

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

   3. associate-*r* 55.5%

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

   4. unpow2 55.5%

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

   5. *-commutative 55.5%

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

   6. unpow2 55.5%

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

   7. swap-sqr 66.4%

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

   8. swap-sqr 79.5%

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

   9. unpow2 79.5%

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

   10. associate-*r* 79.6%

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

   11. *-commutative 79.6%

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

   12. associate-*r* 80.3%

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

9. Simplified 80.3%

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

    1. *-commutative 80.3%

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

    2. *-commutative 80.3%

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

    3. associate-*r* 79.6%

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

    4. pow2 79.6%

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

11. Applied egg-rr 79.6%

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

12. Taylor expanded in x around 0 79.6%

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

    1. associate-*r* 79.6%

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

    2. *-commutative 79.6%

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

    3. associate-*r* 78.7%

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

14. Simplified 78.7%

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

15. Final simplification 78.7%

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

Alternative 10: 78.2% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

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

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = x * (s * c)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. associate-*l* 62.4%

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

   3. associate-*r* 61.3%

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

   4. *-commutative 61.3%

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

   5. unpow2 61.3%

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

   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* 68.3%

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

   8. *-commutative 68.3%

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

   9. unpow2 68.3%

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

3. Simplified 68.3%

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

4. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. unpow2 55.5%

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

   3. *-commutative 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 69.0%

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

   6. unpow2 69.0%

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

   7. associate-/l/ 69.0%

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

   8. unpow2 69.0%

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

   9. swap-sqr 55.6%

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

   10. unpow2 55.6%

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

   11. *-commutative 55.6%

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

   12. associate-*r* 55.5%

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

   13. unpow2 55.5%

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

   14. *-commutative 55.5%

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

   15. unpow2 55.5%

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

6. Simplified 55.5%

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

7. Taylor expanded in c around 0 55.6%

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

   1. *-commutative 55.6%

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

   2. unpow2 55.6%

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

   3. associate-*r* 55.5%

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

   4. unpow2 55.5%

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

   5. *-commutative 55.5%

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

   6. unpow2 55.5%

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

   7. swap-sqr 66.4%

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

   8. swap-sqr 79.5%

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

   9. unpow2 79.5%

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

   10. associate-*r* 79.6%

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

   11. *-commutative 79.6%

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

   12. associate-*r* 80.3%

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

9. Simplified 80.3%

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

    1. *-commutative 80.3%

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

    2. *-commutative 80.3%

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

    3. associate-*r* 79.6%

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

    4. pow2 79.6%

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

11. Applied egg-rr 79.6%

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

12. Final simplification 79.6%

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

Reproduce

herbie shell --seed 2023203 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))