mixedcos

Percentage Accurate: 66.8% → 97.4%

Time: 12.5s

Alternatives: 13

Speedup: 24.1×

• 31.9% 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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

   3. pow2 98.7%

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

   4. pow-flip 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 81.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+143}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot s\right) \cdot {\left(x \cdot c\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 0.42)
   (pow (* c (* x s)) -2.0)
   (if (<= x 2.75e+143)
     (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
     (/ 1.0 (* (* s s) (pow (* x c) 2.0))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.42) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 2.75e+143) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 1.0 / ((s * s) * pow((x * c), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.42d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 2.75d+143) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = 1.0d0 / ((s * s) * ((x * c) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.42) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 2.75e+143) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 1.0 / ((s * s) * Math.pow((x * c), 2.0));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 0.42:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 2.75e+143:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = 1.0 / ((s * s) * math.pow((x * c), 2.0))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 0.42)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 2.75e+143)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = Float64(1.0 / Float64(Float64(s * s) * (Float64(x * c) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.42)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 2.75e+143)
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = 1.0 / ((s * s) * ((x * c) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 0.42], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 2.75e+143], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(s * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s * s), $MachinePrecision] * N[Power[N[(x * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.419999999999999984

   1. Initial program 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r* 62.7%

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

      3. associate-*r* 63.2%

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

      4. unpow2 63.2%

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

      5. unswap-sqr 82.6%

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

      6. unpow2 82.6%

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

      7. swap-sqr 99.6%

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

      8. *-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. *-commutative 99.6%

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

      11. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. *-commutative 99.6%

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

      3. pow2 99.6%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow-prod-up 99.6%

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

      4. pow-prod-down 99.6%

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

      5. pow2 99.6%

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

      6. associate-*r* 97.5%

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

      7. add-sqr-sqrt 50.0%

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

      8. sqrt-prod 84.5%

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

      9. *-commutative 84.5%

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

      10. associate-*r* 84.0%

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

      11. add-sqr-sqrt 40.4%

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

      12. sqrt-prod 73.1%

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

      13. sqrt-prod 73.0%

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

      14. *-commutative 73.0%

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

      15. pow2 73.0%

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

      16. add-sqr-sqrt 73.1%

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

      17. inv-pow 73.1%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0 59.8%

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

      1. unpow2 59.8%

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

      2. associate-/r* 59.8%

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

      3. *-commutative 59.8%

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

      4. unpow2 59.8%

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

      5. unpow2 59.8%

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

      6. swap-sqr 69.9%

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

      7. unpow2 69.9%

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

      8. associate-/l/ 69.9%

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

      9. *-commutative 69.9%

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

      10. unpow2 69.9%

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

      11. swap-sqr 87.9%

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

      12. *-commutative 87.9%

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

      13. associate-*l* 86.1%

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

      14. *-commutative 86.1%

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

      15. associate-*l* 87.7%

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

      16. associate-/l/ 87.7%

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

      17. *-lft-identity 87.7%

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

      18. associate-*l/ 87.7%

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

      19. unpow-1 87.7%

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

      20. unpow-1 87.7%

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

   10. Simplified 87.9%

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

   if 0.419999999999999984 < x < 2.74999999999999985e143

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-*l* 83.7%

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

      3. associate-*r* 86.9%

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

      4. *-commutative 86.9%

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

      5. unpow2 86.9%

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

      6. associate-*r* 87.3%

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

      7. associate-*r* 90.4%

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

      8. *-commutative 90.4%

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

      9. unpow2 90.4%

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

   3. Simplified 90.4%

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

   if 2.74999999999999985e143 < x

   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* 51.8%

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

      3. associate-*r* 51.8%

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

      4. unpow2 51.8%

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

      5. unswap-sqr 66.6%

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

      6. unpow2 66.6%

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

      7. swap-sqr 95.4%

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

      8. *-commutative 95.4%

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

      9. *-commutative 95.4%

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

      10. *-commutative 95.4%

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

      11. *-commutative 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 51.8%

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

      1. associate-/r* 51.8%

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

      2. unpow2 51.8%

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

      3. unpow2 51.8%

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

      4. swap-sqr 60.0%

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

      5. unpow2 60.0%

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

      6. associate-/l/ 60.0%

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

      7. unpow2 60.0%

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

      8. unpow2 60.0%

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

      9. swap-sqr 64.3%

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

      10. associate-*r* 64.4%

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

      11. associate-*r* 64.4%

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

      12. unpow2 64.4%

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

      13. associate-*r* 64.3%

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

      14. *-commutative 64.3%

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

   6. Simplified 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*r* 64.4%

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

      3. *-commutative 64.4%

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

      4. unpow-prod-down 56.1%

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

      5. pow2 56.1%

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

   8. Applied egg-rr 56.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+143}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot s\right) \cdot {\left(x \cdot c\right)}^{2}}\\ \end{array} \]

Alternative 3: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 2.75e+82)
   (/ (cos (* x 2.0)) (* x (* x (* (* s s) (* c c)))))
   (pow (* c (* x s)) -2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 2.75e+82) {
		tmp = cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 2.75d+82) then
        tmp = cos((x * 2.0d0)) / (x * (x * ((s * s) * (c * c))))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 2.75e+82) {
		tmp = Math.cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 2.75e+82:
		tmp = math.cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 2.75e+82)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(Float64(s * s) * Float64(c * c)))));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 2.75e+82)
		tmp = cos((x * 2.0)) / (x * (x * ((s * s) * (c * c))));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 2.75e+82], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 2.74999999999999998e82

   1. Initial program 68.2%

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

      1. associate-*r* 70.3%

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

      2. *-commutative 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-*r* 69.4%

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

      5. *-commutative 69.4%

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

      6. unpow2 69.4%

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

      7. unpow2 69.4%

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

   3. Simplified 69.4%

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

   if 2.74999999999999998e82 < s

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r* 59.9%

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

      3. associate-*r* 61.7%

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

      4. unpow2 61.7%

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

      5. unswap-sqr 74.3%

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

      6. unpow2 74.3%

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

      7. swap-sqr 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. pow2 99.7%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. pow-prod-up 99.6%

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

      4. pow-prod-down 99.7%

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

      5. pow2 99.7%

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

      6. associate-*r* 97.9%

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

      7. add-sqr-sqrt 38.6%

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

      8. sqrt-prod 88.3%

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

      9. *-commutative 88.3%

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

      10. associate-*r* 86.3%

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

      11. add-sqr-sqrt 41.0%

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

      12. sqrt-prod 80.2%

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

      13. sqrt-prod 80.2%

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

      14. *-commutative 80.2%

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

      15. pow2 80.2%

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

      16. add-sqr-sqrt 80.2%

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

      17. inv-pow 80.2%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 59.9%

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

      1. unpow2 59.9%

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

      2. associate-/r* 59.9%

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

      3. *-commutative 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. swap-sqr 82.2%

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

      7. unpow2 82.2%

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

      8. associate-/l/ 82.2%

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

      9. *-commutative 82.2%

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

      10. unpow2 82.2%

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

      11. swap-sqr 91.8%

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

      12. *-commutative 91.8%

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

      13. associate-*l* 90.0%

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

      14. *-commutative 90.0%

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

      15. associate-*l* 90.3%

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

      16. associate-/l/ 90.2%

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

      17. *-lft-identity 90.2%

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

      18. associate-*l/ 90.2%

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

      19. unpow-1 90.2%

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

      20. unpow-1 90.2%

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

   10. Simplified 91.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4: 79.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 8.6e+121)
   (/ (cos (* x 2.0)) (* x (* (* c (* x c)) (* s s))))
   (pow (* c (* x s)) -2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 8.6e+121) {
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 8.6d+121) then
        tmp = cos((x * 2.0d0)) / (x * ((c * (x * c)) * (s * s)))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 8.6e+121) {
		tmp = Math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 8.6e+121:
		tmp = math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 8.6e+121)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 8.6e+121)
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 8.6e+121], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 8.5999999999999994e121

   1. Initial program 68.2%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*r* 71.5%

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

      4. unpow2 71.5%

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

      5. unpow2 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in c around 0 71.5%

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

      1. unpow2 71.5%

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

      2. associate-*l* 80.4%

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

   6. Simplified 80.4%

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

   if 8.5999999999999994e121 < s

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*r* 57.3%

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

      3. associate-*r* 57.1%

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

      4. unpow2 57.1%

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

      5. unswap-sqr 71.2%

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

      6. unpow2 71.2%

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

      7. swap-sqr 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. pow2 99.7%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. pow-prod-up 99.7%

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

      4. pow-prod-down 99.7%

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

      5. pow2 99.7%

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

      6. associate-*r* 99.8%

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

      7. add-sqr-sqrt 37.8%

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

      8. sqrt-prod 92.1%

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

      9. *-commutative 92.1%

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

      10. associate-*r* 89.5%

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

      11. add-sqr-sqrt 46.0%

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

      12. sqrt-prod 84.1%

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

      13. sqrt-prod 84.1%

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

      14. *-commutative 84.1%

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

      15. pow2 84.1%

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

      16. add-sqr-sqrt 84.2%

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

      17. inv-pow 84.2%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 57.3%

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

      1. unpow2 57.3%

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

      2. associate-/r* 57.3%

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

      3. *-commutative 57.3%

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

      4. unpow2 57.3%

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

      5. unpow2 57.3%

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

      6. swap-sqr 86.7%

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

      7. unpow2 86.7%

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

      8. associate-/l/ 86.7%

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

      9. *-commutative 86.7%

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

      10. unpow2 86.7%

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

      11. swap-sqr 96.1%

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

      12. *-commutative 96.1%

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

      13. associate-*l* 93.6%

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

      14. *-commutative 93.6%

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

      15. associate-*l* 93.6%

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

      16. associate-/l/ 93.5%

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

      17. *-lft-identity 93.5%

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

      18. associate-*l/ 93.5%

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

      19. unpow-1 93.5%

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

      20. unpow-1 93.5%

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

   10. Simplified 96.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 5: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $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 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. associate-/r* 98.9%

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

   2. div-inv 98.9%

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

   3. *-commutative 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 6: 94.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in s around 0 97.2%

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

5. Final simplification 97.2%

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

Alternative 7: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 8: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

   3. pow2 98.7%

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

   4. pow-flip 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. pow-prod-up 98.8%

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

   4. pow-prod-down 98.7%

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

   5. pow2 98.7%

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

   6. associate-*r* 97.9%

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

   7. add-sqr-sqrt 51.3%

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

   8. sqrt-prod 86.3%

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

   9. *-commutative 86.3%

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

   10. associate-*r* 84.6%

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

   11. add-sqr-sqrt 41.2%

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

   12. sqrt-prod 75.2%

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

   13. sqrt-prod 75.1%

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

   14. *-commutative 75.1%

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

   15. pow2 75.1%

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

   16. add-sqr-sqrt 75.2%

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

   17. inv-pow 75.2%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 9: 78.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * (x * s)) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in x around 0 58.6%

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

   1. associate-/r* 58.6%

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

   2. unpow2 58.6%

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

   3. unpow2 58.6%

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

   4. swap-sqr 67.3%

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

   5. unpow2 67.3%

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

   6. associate-/l/ 67.3%

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

   7. unpow2 67.3%

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

   8. unpow2 67.3%

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

   9. swap-sqr 81.5%

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

   10. associate-*r* 81.6%

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

   11. associate-*r* 82.8%

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

   12. unpow2 82.8%

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

   13. associate-*r* 81.5%

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

   14. *-commutative 81.5%

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

6. Simplified 81.5%

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

7. Final simplification 81.5%

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

Alternative 10: 78.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

   3. pow2 98.7%

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

   4. pow-flip 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. pow-prod-up 98.8%

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

   4. pow-prod-down 98.7%

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

   5. pow2 98.7%

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

   6. associate-*r* 97.9%

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

   7. add-sqr-sqrt 51.3%

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

   8. sqrt-prod 86.3%

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

   9. *-commutative 86.3%

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

   10. associate-*r* 84.6%

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

   11. add-sqr-sqrt 41.2%

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

   12. sqrt-prod 75.2%

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

   13. sqrt-prod 75.1%

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

   14. *-commutative 75.1%

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

   15. pow2 75.1%

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

   16. add-sqr-sqrt 75.2%

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

   17. inv-pow 75.2%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in x around 0 58.6%

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

   1. unpow2 58.6%

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

   2. associate-/r* 58.6%

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

   3. *-commutative 58.6%

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

   4. unpow2 58.6%

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

   5. unpow2 58.6%

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

   6. swap-sqr 67.3%

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

   7. unpow2 67.3%

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

   8. associate-/l/ 67.3%

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

   9. *-commutative 67.3%

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

   10. unpow2 67.3%

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

   11. swap-sqr 81.5%

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

   12. *-commutative 81.5%

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

   13. associate-*l* 80.2%

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

   14. *-commutative 80.2%

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

   15. associate-*l* 81.4%

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

   16. associate-/l/ 81.4%

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

   17. *-lft-identity 81.4%

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

   18. associate-*l/ 81.4%

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

   19. unpow-1 81.4%

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

   20. unpow-1 81.4%

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

10. Simplified 81.6%

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

11. Final simplification 81.6%

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

Alternative 11: 55.8% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if s < 1.35000000000000003e154

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 64.1%

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

      3. associate-*r* 65.5%

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

      4. unpow2 65.5%

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

      5. unswap-sqr 82.1%

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

      6. unpow2 82.1%

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

      7. swap-sqr 98.6%

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

      8. *-commutative 98.6%

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

      9. *-commutative 98.6%

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

      10. *-commutative 98.6%

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

      11. *-commutative 98.6%

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

   3. Simplified 98.6%

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

      1. div-inv 98.6%

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

      2. *-commutative 98.6%

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

      3. pow2 98.6%

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

      4. pow-flip 98.8%

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

      5. metadata-eval 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow-prod-up 98.7%

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

      4. pow-prod-down 98.6%

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

      5. pow2 98.6%

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

      6. associate-*r* 97.6%

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

      7. add-sqr-sqrt 53.1%

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

      8. sqrt-prod 85.1%

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

      9. *-commutative 85.1%

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

      10. associate-*r* 83.6%

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

      11. add-sqr-sqrt 40.6%

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

      12. sqrt-prod 73.7%

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

      13. sqrt-prod 73.6%

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

      14. *-commutative 73.6%

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

      15. pow2 73.6%

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

      16. add-sqr-sqrt 73.7%

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

      17. inv-pow 73.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around 0 58.2%

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

      1. unpow2 58.2%

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

      2. associate-*r* 58.2%

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

      3. *-commutative 58.2%

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

      4. associate-*r* 58.8%

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

      5. unpow2 58.8%

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

      6. *-commutative 58.8%

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

      7. unpow2 58.8%

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

   10. Simplified 58.8%

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

   if 1.35000000000000003e154 < s

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 60.9%

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

      3. associate-*r* 60.6%

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

      4. unpow2 60.6%

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

      5. unswap-sqr 76.4%

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

      6. unpow2 76.4%

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

      7. swap-sqr 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

      11. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 60.6%

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

      1. associate-/r* 60.6%

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

      2. unpow2 60.6%

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

      3. unpow2 60.6%

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

      4. unpow2 60.6%

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

      5. associate-*r/ 60.6%

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

      6. metadata-eval 60.6%

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

      7. unpow2 60.6%

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

      8. unpow2 60.6%

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

   6. Simplified 60.6%

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

   7. Taylor expanded in x around inf 73.6%

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

      1. unpow2 73.6%

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

      2. *-commutative 73.6%

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

      3. unpow2 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 60.7%

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

Alternative 12: 78.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = x * (s * c)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in x around 0 58.6%

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

   1. associate-/r* 58.6%

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

   2. unpow2 58.6%

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

   3. unpow2 58.6%

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

   4. swap-sqr 67.3%

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

   5. unpow2 67.3%

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

   6. associate-/l/ 67.3%

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

   7. unpow2 67.3%

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

   8. unpow2 67.3%

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

   9. swap-sqr 81.5%

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

   10. associate-*r* 81.6%

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

   11. associate-*r* 82.8%

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

   12. unpow2 82.8%

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

   13. associate-*r* 81.5%

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

   14. *-commutative 81.5%

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

6. Simplified 81.5%

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

   1. unpow2 81.5%

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

   2. associate-*r* 80.2%

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

   3. associate-*r* 81.4%

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

8. Applied egg-rr 81.4%

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

9. Final simplification 81.4%

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

Alternative 13: 28.1% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*r* 63.7%

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

   3. associate-*r* 64.8%

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

   4. unpow2 64.8%

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

   5. unswap-sqr 81.4%

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

   6. unpow2 81.4%

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

   7. swap-sqr 98.7%

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

   8. *-commutative 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in x around 0 32.0%

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

   1. associate-/r* 32.0%

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

   2. unpow2 32.0%

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

   3. unpow2 32.0%

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

   4. unpow2 32.0%

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

   5. associate-*r/ 32.0%

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

   6. metadata-eval 32.0%

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

   7. unpow2 32.0%

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

   8. unpow2 32.0%

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

6. Simplified 32.0%

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

7. Taylor expanded in x around inf 27.0%

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

   1. unpow2 27.0%

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

   2. *-commutative 27.0%

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

   3. unpow2 27.0%

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

9. Simplified 27.0%

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

10. Final simplification 27.0%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))