mixedcos

Percentage Accurate: 66.0% → 97.4%

Time: 12.8s

Alternatives: 13

Speedup: 24.1×

• Unsound rule application detected in e-graph. Results may not be sound.

• 31.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% 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: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return ((1.0 / x) / (c * s)) * (cos((x * 2.0)) / (x * (c * 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 = ((1.0d0 / x) / (c * s)) * (cos((x * 2.0d0)) / (x * (c * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

   3. *-commutative 64.5%

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

   4. associate-*r* 63.0%

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

   5. *-commutative 63.0%

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

   6. unpow2 63.0%

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

   7. unpow2 63.0%

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

3. Simplified 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. associate-*r* 64.1%

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

   3. *-commutative 64.1%

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

   4. add-sqr-sqrt 64.1%

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

   5. times-frac 64.0%

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

5. Applied egg-rr 98.2%

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

6. Taylor expanded in x around 0 95.4%

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

   1. *-commutative 95.4%

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

   2. *-commutative 95.4%

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

   3. associate-*r* 98.2%

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

   4. associate-/r* 98.2%

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

   5. *-commutative 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 83.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot t_0}\\ \mathbf{elif}\;x \leq -0.013 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))))
   (if (<= x -8.8e+154)
     (/ 1.0 (* (* s (* x c)) t_0))
     (if (or (<= x -0.013) (not (<= x 0.62)))
       (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
       (pow t_0 -2.0)))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= -8.8e+154) {
		tmp = 1.0 / ((s * (x * c)) * t_0);
	} else if ((x <= -0.013) || !(x <= 0.62)) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = pow(t_0, -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) :: t_0
    real(8) :: tmp
    t_0 = x * (c * s)
    if (x <= (-8.8d+154)) then
        tmp = 1.0d0 / ((s * (x * c)) * t_0)
    else if ((x <= (-0.013d0)) .or. (.not. (x <= 0.62d0))) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = t_0 ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= -8.8e+154) {
		tmp = 1.0 / ((s * (x * c)) * t_0);
	} else if ((x <= -0.013) || !(x <= 0.62)) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = Math.pow(t_0, -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	tmp = 0
	if x <= -8.8e+154:
		tmp = 1.0 / ((s * (x * c)) * t_0)
	elif (x <= -0.013) or not (x <= 0.62):
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = math.pow(t_0, -2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (x <= -8.8e+154)
		tmp = Float64(1.0 / Float64(Float64(s * Float64(x * c)) * t_0));
	elseif ((x <= -0.013) || !(x <= 0.62))
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = t_0 ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	tmp = 0.0;
	if (x <= -8.8e+154)
		tmp = 1.0 / ((s * (x * c)) * t_0);
	elseif ((x <= -0.013) || ~((x <= 0.62)))
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = t_0 ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.8e+154], N[(1.0 / N[(N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -0.013], N[Not[LessEqual[x, 0.62]], $MachinePrecision]], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(s * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[t$95$0, -2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000004e154

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-*r* 63.4%

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

      3. associate-*r* 63.4%

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

      4. unpow2 63.4%

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

      5. unswap-sqr 84.9%

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

      6. unpow2 84.9%

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

      7. swap-sqr 94.7%

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

      8. *-commutative 94.7%

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

      9. *-commutative 94.7%

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

      10. *-commutative 94.7%

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

      11. *-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x around 0 63.4%

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

      1. unpow2 63.4%

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

      2. unpow2 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-*r* 63.4%

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

      5. unpow2 63.4%

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

      6. unpow2 63.4%

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

      7. *-commutative 63.4%

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

      8. unpow2 63.4%

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

      9. swap-sqr 79.1%

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

      10. unpow2 79.1%

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

      11. associate-*r* 79.6%

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

      12. *-commutative 79.6%

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

      13. *-commutative 79.6%

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

      14. associate-*r* 79.1%

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

      15. unpow2 79.1%

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

      16. swap-sqr 79.9%

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

      17. unpow2 79.9%

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

   6. Simplified 79.9%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. pow2 79.9%

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

      5. *-commutative 79.9%

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

      6. *-commutative 79.9%

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

   8. Applied egg-rr 79.9%

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

   9. Taylor expanded in x around 0 79.9%

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

   if -8.8000000000000004e154 < x < -0.0129999999999999994 or 0.619999999999999996 < x

   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* 59.2%

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

      3. associate-*r* 57.2%

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

      4. *-commutative 57.2%

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

      5. unpow2 57.2%

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

      6. associate-*r* 62.3%

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

      7. associate-*r* 64.2%

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

      8. *-commutative 64.2%

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

      9. unpow2 64.2%

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

   3. Simplified 64.2%

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

   if -0.0129999999999999994 < x < 0.619999999999999996

   1. Initial program 61.3%

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

      1. *-commutative 61.3%

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

      2. associate-*r* 56.4%

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

      3. associate-*r* 57.0%

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

      4. unpow2 57.0%

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

      5. unswap-sqr 70.3%

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

      6. unpow2 70.3%

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

      7. swap-sqr 97.1%

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

      8. *-commutative 97.1%

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

      9. *-commutative 97.1%

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

      10. *-commutative 97.1%

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

      11. *-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. unpow2 56.4%

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

   6. Simplified 56.4%

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

      1. add-sqr-sqrt 56.4%

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

      2. pow2 56.4%

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

      3. sqrt-div 56.4%

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

      4. metadata-eval 56.4%

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

      5. associate-*r* 61.3%

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

      6. associate-*r* 69.5%

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

      7. *-commutative 69.5%

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

      8. *-commutative 69.5%

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

      9. sqrt-prod 69.5%

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

      10. associate-*r* 72.5%

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

      11. *-commutative 72.5%

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

      12. sqrt-prod 51.5%

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

      13. add-sqr-sqrt 77.7%

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

      14. *-commutative 77.7%

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

      15. sqrt-prod 53.0%

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

      16. add-sqr-sqrt 94.8%

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

      17. associate-*r* 97.6%

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

   8. Applied egg-rr 97.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq -0.013 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 3: 88.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+32} \lor \neg \left(x \leq 0.64\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (or (<= x -1.5e+32) (not (<= x 0.64)))
   (/ (cos (* x 2.0)) (* x (* c (* c (* s (* x s))))))
   (pow (* x (* c s)) -2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if ((x <= -1.5e+32) || !(x <= 0.64)) {
		tmp = cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	} else {
		tmp = pow((x * (c * 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 ((x <= (-1.5d+32)) .or. (.not. (x <= 0.64d0))) then
        tmp = cos((x * 2.0d0)) / (x * (c * (c * (s * (x * s)))))
    else
        tmp = (x * (c * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((x <= -1.5e+32) || !(x <= 0.64)) {
		tmp = Math.cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	} else {
		tmp = Math.pow((x * (c * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (x <= -1.5e+32) or not (x <= 0.64):
		tmp = math.cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))))
	else:
		tmp = math.pow((x * (c * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if ((x <= -1.5e+32) || !(x <= 0.64))
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(c * Float64(c * Float64(s * Float64(x * s))))));
	else
		tmp = Float64(x * Float64(c * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((x <= -1.5e+32) || ~((x <= 0.64)))
		tmp = cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	else
		tmp = (x * (c * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[Or[LessEqual[x, -1.5e+32], N[Not[LessEqual[x, 0.64]], $MachinePrecision]], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e32 or 0.640000000000000013 < x

   1. Initial program 66.6%

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

      1. associate-*r* 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r* 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%

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

      1. *-commutative 67.0%

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

      2. unpow2 67.0%

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

      3. associate-*r* 70.7%

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

      4. *-commutative 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in c around 0 67.0%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*r* 67.9%

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

      4. unpow2 67.9%

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

      5. associate-*l* 74.1%

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

      6. unpow2 74.1%

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

      7. associate-*l* 81.3%

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

   9. Simplified 81.3%

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

   if -1.5e32 < x < 0.640000000000000013

   1. Initial program 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*r* 56.3%

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

      3. associate-*r* 56.9%

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

      4. unpow2 56.9%

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

      5. unswap-sqr 69.8%

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

      6. unpow2 69.8%

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

      7. swap-sqr 96.7%

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

      8. *-commutative 96.7%

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

      9. *-commutative 96.7%

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

      10. *-commutative 96.7%

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

      11. *-commutative 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around 0 56.3%

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

      1. unpow2 56.3%

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

      2. unpow2 56.3%

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

      3. unpow2 56.3%

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

   6. Simplified 56.3%

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

      1. add-sqr-sqrt 56.2%

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

      2. pow2 56.2%

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

      3. sqrt-div 56.2%

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

      4. metadata-eval 56.2%

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

      5. associate-*r* 61.0%

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

      6. associate-*r* 69.0%

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

      7. *-commutative 69.0%

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

      8. *-commutative 69.0%

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

      9. sqrt-prod 69.0%

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

      10. associate-*r* 72.0%

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

      11. *-commutative 72.0%

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

      12. sqrt-prod 50.8%

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

      13. add-sqr-sqrt 77.0%

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

      14. *-commutative 77.0%

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

      15. sqrt-prod 52.6%

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

      16. add-sqr-sqrt 94.0%

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

      17. associate-*r* 96.7%

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

   8. Applied egg-rr 96.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+32} \lor \neg \left(x \leq 0.64\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4: 95.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;s \leq 8.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= s 8.5e+184)
     (/ (cos (* x 2.0)) (* t_0 (* s (* x c))))
     (pow t_0 -2.0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (s <= 8.5e+184) {
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	} else {
		tmp = pow(t_0, -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) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (s <= 8.5d+184) then
        tmp = cos((x * 2.0d0)) / (t_0 * (s * (x * c)))
    else
        tmp = t_0 ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (s <= 8.5e+184) {
		tmp = Math.cos((x * 2.0)) / (t_0 * (s * (x * c)));
	} else {
		tmp = Math.pow(t_0, -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if s <= 8.5e+184:
		tmp = math.cos((x * 2.0)) / (t_0 * (s * (x * c)))
	else:
		tmp = math.pow(t_0, -2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (s <= 8.5e+184)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * Float64(s * Float64(x * c))));
	else
		tmp = t_0 ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (s <= 8.5e+184)
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	else
		tmp = t_0 ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 8.5e+184], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[t$95$0, -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 8.50000000000000043e184

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-*r* 58.0%

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

      3. associate-*r* 57.6%

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

      4. unpow2 57.6%

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

      5. unswap-sqr 72.8%

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

      6. unpow2 72.8%

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

      7. swap-sqr 96.5%

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

      8. *-commutative 96.5%

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

      9. *-commutative 96.5%

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

      10. *-commutative 96.5%

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

      11. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in s around 0 93.6%

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

   if 8.50000000000000043e184 < s

   1. Initial program 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r* 63.8%

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

      5. *-commutative 63.8%

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

      6. unpow2 63.8%

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

      7. unpow2 63.8%

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

   3. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. associate-*r* 63.4%

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

      3. *-commutative 63.4%

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

      4. add-sqr-sqrt 63.4%

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

      5. times-frac 63.4%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

      3. *-commutative 59.8%

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

      4. unpow2 59.8%

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

      5. *-commutative 59.8%

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

      6. unpow2 59.8%

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

      7. unpow2 59.8%

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

      8. swap-sqr 82.1%

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

      9. swap-sqr 99.9%

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

      10. associate-/r* 99.9%

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

      11. *-lft-identity 99.9%

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

      12. associate-*l/ 99.8%

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

      13. unpow-1 99.8%

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

      14. unpow-1 99.8%

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

      15. pow-sqr 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-*r* 99.9%

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

      18. *-commutative 99.9%

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

      19. *-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 8.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 5: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 1.28 \cdot 10^{+185}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= s 1.28e+185)
     (/ (cos (* x 2.0)) (* t_0 t_0))
     (pow (* c (* x s)) -2.0))))

C

double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (s <= 1.28e+185) {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (s <= 1.28d+185) then
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    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 t_0 = s * (x * c);
	double tmp;
	if (s <= 1.28e+185) {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if s <= 1.28e+185:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (s <= 1.28e+185)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (s <= 1.28e+185)
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 1.28e+185], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.27999999999999993e185

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-*r* 58.0%

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

      3. associate-*r* 57.6%

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

      4. unpow2 57.6%

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

      5. unswap-sqr 72.8%

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

      6. unpow2 72.8%

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

      7. swap-sqr 96.5%

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

      8. *-commutative 96.5%

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

      9. *-commutative 96.5%

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

      10. *-commutative 96.5%

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

      11. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   if 1.27999999999999993e185 < s

   1. Initial program 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r* 63.8%

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

      5. *-commutative 63.8%

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

      6. unpow2 63.8%

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

      7. unpow2 63.8%

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

   3. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. associate-*r* 63.4%

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

      3. *-commutative 63.4%

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

      4. add-sqr-sqrt 63.4%

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

      5. times-frac 63.4%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

      3. *-commutative 59.8%

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

      4. unpow2 59.8%

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

      5. *-commutative 59.8%

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

      6. unpow2 59.8%

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

      7. unpow2 59.8%

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

      8. swap-sqr 82.1%

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

      9. swap-sqr 99.9%

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

      10. associate-/r* 99.9%

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

      11. *-lft-identity 99.9%

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

      12. associate-*l/ 99.8%

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

      13. unpow-1 99.8%

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

      14. unpow-1 99.8%

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

      15. pow-sqr 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-*r* 99.9%

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

      18. *-commutative 99.9%

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

      19. *-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.28 \cdot 10^{+185}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 6: 95.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return (1.0 / (x * (c * s))) * (cos((x * 2.0)) / (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 = (1.0d0 / (x * (c * s))) * (cos((x * 2.0d0)) / (c * (x * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

   3. *-commutative 64.5%

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

   4. associate-*r* 63.0%

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

   5. *-commutative 63.0%

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

   6. unpow2 63.0%

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

   7. unpow2 63.0%

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

3. Simplified 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. associate-*r* 64.1%

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

   3. *-commutative 64.1%

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

   4. add-sqr-sqrt 64.1%

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

   5. times-frac 64.0%

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

5. Applied egg-rr 98.2%

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

6. Taylor expanded in x around 0 95.4%

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

7. Final simplification 95.4%

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

Alternative 7: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \frac{\cos \left(x \cdot 2\right)}{t_0} \cdot \frac{1}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s)))) (* (/ (cos (* x 2.0)) t_0) (/ 1.0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	return (cos((x * 2.0)) / t_0) * (1.0 / 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 * (c * s)
    code = (cos((x * 2.0d0)) / t_0) * (1.0d0 / t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	return (Math.cos((x * 2.0)) / t_0) * (1.0 / t_0);
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	return (math.cos((x * 2.0)) / t_0) * (1.0 / t_0)

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	return Float64(Float64(cos(Float64(x * 2.0)) / t_0) * Float64(1.0 / t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (c * s);
	tmp = (cos((x * 2.0)) / t_0) * (1.0 / t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.4%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

   3. *-commutative 64.5%

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

   4. associate-*r* 63.0%

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

   5. *-commutative 63.0%

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

   6. unpow2 63.0%

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

   7. unpow2 63.0%

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

3. Simplified 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. associate-*r* 64.1%

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

   3. *-commutative 64.1%

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

   4. add-sqr-sqrt 64.1%

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

   5. times-frac 64.0%

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

5. Applied egg-rr 98.2%

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

6. Final simplification 98.2%

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

Alternative 8: 78.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return pow((x * (c * 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 = (x * (c * s)) ** (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.2%

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

   3. associate-*r* 57.8%

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

   4. unpow2 57.8%

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

   5. unswap-sqr 72.2%

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

   6. unpow2 72.2%

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

   7. swap-sqr 96.3%

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

   8. *-commutative 96.3%

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

   9. *-commutative 96.3%

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

   10. *-commutative 96.3%

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

   11. *-commutative 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. unpow2 52.8%

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

   3. unpow2 52.8%

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

6. Simplified 52.8%

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

   1. add-sqr-sqrt 52.7%

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

   2. pow2 52.7%

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

   3. sqrt-div 52.7%

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

   4. metadata-eval 52.7%

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

   5. associate-*r* 56.7%

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

   6. associate-*r* 62.1%

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

   7. *-commutative 62.1%

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

   8. *-commutative 62.1%

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

   9. sqrt-prod 62.1%

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

   10. associate-*r* 63.8%

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

   11. *-commutative 63.8%

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

   12. sqrt-prod 39.4%

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

   13. add-sqr-sqrt 66.9%

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

   14. *-commutative 66.9%

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

   15. sqrt-prod 39.3%

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

   16. add-sqr-sqrt 77.9%

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

   17. associate-*r* 79.3%

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

8. Applied egg-rr 79.3%

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

9. Final simplification 79.3%

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

Alternative 9: 56.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.35e+154)
   (/ 1.0 (* (* c c) (* (* x x) (* s s))))
   (/ -2.0 (* c (* c (* s s))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.35e+154) {
		tmp = 1.0 / ((c * c) * ((x * x) * (s * s)));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	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 <= 1.35d+154) then
        tmp = 1.0d0 / ((c * c) * ((x * x) * (s * s)))
    else
        tmp = (-2.0d0) / (c * (c * (s * s)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.35e+154) {
		tmp = 1.0 / ((c * c) * ((x * x) * (s * s)));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.35e+154:
		tmp = 1.0 / ((c * c) * ((x * x) * (s * s)))
	else:
		tmp = -2.0 / (c * (c * (s * s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(x * x) * Float64(s * s))));
	else
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 1.35e+154)
		tmp = 1.0 / ((c * c) * ((x * x) * (s * s)));
	else
		tmp = -2.0 / (c * (c * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.35e+154], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.35000000000000003e154

   1. Initial program 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 58.2%

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

      3. associate-*r* 57.8%

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

      4. unpow2 57.8%

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

      5. unswap-sqr 73.4%

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

      6. unpow2 73.4%

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

      7. swap-sqr 96.4%

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

      8. *-commutative 96.4%

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

      9. *-commutative 96.4%

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

      10. *-commutative 96.4%

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

      11. *-commutative 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in x around 0 52.0%

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

      1. unpow2 52.0%

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

      2. unpow2 52.0%

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

      3. unpow2 52.0%

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

   6. Simplified 52.0%

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

   if 1.35000000000000003e154 < s

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*r* 58.0%

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

      3. associate-*r* 57.7%

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

      4. unpow2 57.7%

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

      5. unswap-sqr 63.8%

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

      6. unpow2 63.8%

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

      7. swap-sqr 95.3%

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

      8. *-commutative 95.3%

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

      9. *-commutative 95.3%

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

      10. *-commutative 95.3%

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

      11. *-commutative 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in x around 0 57.7%

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

      1. unpow2 57.7%

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

      2. unpow2 57.7%

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

      3. associate-*r* 58.0%

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

      4. *-commutative 58.0%

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

      5. associate-*r* 58.0%

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

      6. unpow2 58.0%

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

      7. unpow2 58.0%

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

      8. unpow2 58.0%

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

      9. unpow2 58.0%

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

      10. unpow2 58.0%

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

      11. associate-*r/ 58.0%

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

      12. metadata-eval 58.0%

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

      13. unpow2 58.0%

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

      14. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in x around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. unpow2 61.4%

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

      3. associate-*r* 64.6%

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

      4. unpow2 64.6%

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

   9. Simplified 64.6%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 10: 77.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.2%

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

   3. associate-*r* 57.8%

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

   4. unpow2 57.8%

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

   5. unswap-sqr 72.2%

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

   6. unpow2 72.2%

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

   7. swap-sqr 96.3%

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

   8. *-commutative 96.3%

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

   9. *-commutative 96.3%

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

   10. *-commutative 96.3%

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

   11. *-commutative 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. unpow2 52.8%

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

   3. *-commutative 52.8%

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

   4. associate-*r* 53.0%

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

   5. unpow2 53.0%

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

   6. unpow2 53.0%

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

   7. *-commutative 53.0%

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

   8. unpow2 53.0%

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

   9. swap-sqr 63.0%

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

   10. unpow2 63.0%

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

   11. associate-*r* 70.5%

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

   12. *-commutative 70.5%

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

   13. *-commutative 70.5%

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

   14. associate-*r* 63.0%

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

   15. unpow2 63.0%

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

   16. swap-sqr 78.8%

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

   17. unpow2 78.8%

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

6. Simplified 77.8%

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

   1. associate-*r* 79.3%

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

   2. *-commutative 79.3%

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

   3. *-commutative 79.3%

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

   4. pow2 79.3%

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

   5. *-commutative 79.3%

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

   6. *-commutative 79.3%

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

8. Applied egg-rr 79.3%

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

9. Taylor expanded in x around 0 78.1%

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

    1. associate-*r* 79.3%

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

    2. *-commutative 79.3%

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

    3. associate-*r* 77.1%

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

11. Simplified 77.1%

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

12. Final simplification 77.1%

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

Alternative 11: 78.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	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 * (c * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.2%

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

   3. associate-*r* 57.8%

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

   4. unpow2 57.8%

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

   5. unswap-sqr 72.2%

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

   6. unpow2 72.2%

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

   7. swap-sqr 96.3%

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

   8. *-commutative 96.3%

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

   9. *-commutative 96.3%

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

   10. *-commutative 96.3%

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

   11. *-commutative 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in x around 0 52.8%

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

   1. unpow2 52.8%

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

   2. unpow2 52.8%

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

   3. *-commutative 52.8%

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

   4. associate-*r* 53.0%

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

   5. unpow2 53.0%

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

   6. unpow2 53.0%

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

   7. *-commutative 53.0%

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

   8. unpow2 53.0%

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

   9. swap-sqr 63.0%

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

   10. unpow2 63.0%

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

   11. associate-*r* 70.5%

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

   12. *-commutative 70.5%

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

   13. *-commutative 70.5%

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

   14. associate-*r* 63.0%

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

   15. unpow2 63.0%

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

   16. swap-sqr 78.8%

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

   17. unpow2 78.8%

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

6. Simplified 77.8%

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

   1. associate-*r* 79.3%

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

   2. *-commutative 79.3%

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

   3. *-commutative 79.3%

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

   4. pow2 79.3%

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

   5. *-commutative 79.3%

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

   6. *-commutative 79.3%

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

8. Applied egg-rr 79.3%

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

9. Final simplification 79.3%

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

Alternative 12: 28.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return -2.0 / (c * (c * (s * 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 = (-2.0d0) / (c * (c * (s * s)))
end function

Java

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

Python

def code(x, c, s):
	return -2.0 / (c * (c * (s * s)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.2%

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

   3. associate-*r* 57.8%

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

   4. unpow2 57.8%

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

   5. unswap-sqr 72.2%

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

   6. unpow2 72.2%

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

   7. swap-sqr 96.3%

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

   8. *-commutative 96.3%

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

   9. *-commutative 96.3%

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

   10. *-commutative 96.3%

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

   11. *-commutative 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in x around 0 32.9%

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

   1. unpow2 32.9%

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

   2. unpow2 32.9%

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

   3. associate-*r* 32.3%

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

   4. *-commutative 32.3%

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

   5. associate-*r* 32.6%

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

   6. unpow2 32.6%

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

   7. unpow2 32.6%

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

   8. unpow2 32.6%

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

   9. unpow2 32.6%

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

   10. unpow2 32.6%

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

   11. associate-*r/ 32.6%

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

   12. metadata-eval 32.6%

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

   13. unpow2 32.6%

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

   14. unpow2 32.6%

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

6. Simplified 32.6%

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

7. Taylor expanded in x around inf 28.1%

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

   1. *-commutative 28.1%

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

   2. unpow2 28.1%

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

   3. associate-*r* 27.2%

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

   4. unpow2 27.2%

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

9. Simplified 27.2%

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

10. Final simplification 27.2%

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

Alternative 13: 27.7% accurate, 34.8× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return -2.0 / ((c * c) * (s * 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 = (-2.0d0) / ((c * c) * (s * s))
end function

Java

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

Python

def code(x, c, s):
	return -2.0 / ((c * c) * (s * s))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-commutative 63.4%

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

   2. associate-*r* 58.2%

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

   3. associate-*r* 57.8%

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

   4. unpow2 57.8%

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

   5. unswap-sqr 72.2%

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

   6. unpow2 72.2%

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

   7. swap-sqr 96.3%

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

   8. *-commutative 96.3%

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

   9. *-commutative 96.3%

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

   10. *-commutative 96.3%

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

   11. *-commutative 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in x around 0 32.9%

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

   1. unpow2 32.9%

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

   2. unpow2 32.9%

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

   3. associate-*r* 32.3%

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

   4. *-commutative 32.3%

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

   5. associate-*r* 32.6%

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

   6. unpow2 32.6%

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

   7. unpow2 32.6%

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

   8. unpow2 32.6%

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

   9. unpow2 32.6%

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

   10. unpow2 32.6%

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

   11. associate-*r/ 32.6%

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

   12. metadata-eval 32.6%

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

   13. unpow2 32.6%

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

   14. unpow2 32.6%

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

6. Simplified 32.6%

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

7. Taylor expanded in x around inf 28.1%

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

   1. *-commutative 28.1%

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

   2. unpow2 28.1%

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

   3. unpow2 28.1%

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

9. Simplified 28.1%

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

10. Final simplification 28.1%

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))