mixedcos

Percentage Accurate: 67.6% → 97.3%

Time: 13.6s

Alternatives: 14

Speedup: 20.9×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.6% 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: 97.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = 1.0 / ((s * c) * x);
	return (t_0 * t_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
    real(8) :: t_0
    t_0 = 1.0d0 / ((s * c) * x)
    code = (t_0 * t_0) * cos((x * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in s around 0 66.8%

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

   1. *-commutative 66.8%

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

   2. unpow2 66.8%

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

   3. unpow2 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 81.8%

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

   6. unpow2 81.8%

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

   7. swap-sqr 98.2%

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

   8. *-commutative 98.2%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

   11. associate-*r* 97.3%

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

   12. unpow2 97.3%

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

   13. associate-*r* 98.2%

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

   14. *-commutative 98.2%

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

6. Simplified 98.2%

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

   1. unpow2 98.2%

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

8. Applied egg-rr 98.2%

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

9. Taylor expanded in x around inf 66.1%

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

    1. *-commutative 66.1%

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

    2. associate-*r* 66.3%

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

    3. unpow2 66.3%

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

    4. unpow2 66.3%

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

    5. unswap-sqr 80.0%

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

    6. unpow2 80.0%

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

    7. swap-sqr 98.5%

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

    8. associate-/r* 98.5%

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

    9. *-commutative 98.5%

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

    10. associate-*r* 97.2%

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

    11. *-lft-identity 97.2%

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

    12. times-frac 95.2%

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

    13. associate-/l/ 95.1%

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

    14. times-frac 94.9%

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

    15. associate-*l/ 98.2%

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

    16. associate-/r* 98.1%

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

11. Simplified 97.3%

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

    1. metadata-eval 97.3%

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

    2. pow-prod-up 97.3%

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

    3. unpow-1 97.3%

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

    4. associate-*r* 96.3%

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

    5. unpow-1 96.3%

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

    6. associate-*r* 98.5%

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

13. Applied egg-rr 98.5%

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

14. Final simplification 98.5%

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

Alternative 2: 82.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left(s \cdot c\right) \cdot x}\\ \mathbf{if}\;c \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-160} \lor \neg \left(c \leq 5.3 \cdot 10^{-160}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (* s c) x))))
   (if (<= c -1.4e+154)
     (* t_0 t_0)
     (if (or (<= c -1.7e-160) (not (<= c 5.3e-160)))
       (/ (cos (* x 2.0)) (* x (* s (* (* s x) (* c c)))))
       (pow (* s (* c x)) -2.0)))))

C

double code(double x, double c, double s) {
	double t_0 = 1.0 / ((s * c) * x);
	double tmp;
	if (c <= -1.4e+154) {
		tmp = t_0 * t_0;
	} else if ((c <= -1.7e-160) || !(c <= 5.3e-160)) {
		tmp = cos((x * 2.0)) / (x * (s * ((s * x) * (c * c))));
	} else {
		tmp = pow((s * (c * x)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / ((s * c) * x)
    if (c <= (-1.4d+154)) then
        tmp = t_0 * t_0
    else if ((c <= (-1.7d-160)) .or. (.not. (c <= 5.3d-160))) then
        tmp = cos((x * 2.0d0)) / (x * (s * ((s * x) * (c * c))))
    else
        tmp = (s * (c * x)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = 1.0 / ((s * c) * x);
	double tmp;
	if (c <= -1.4e+154) {
		tmp = t_0 * t_0;
	} else if ((c <= -1.7e-160) || !(c <= 5.3e-160)) {
		tmp = Math.cos((x * 2.0)) / (x * (s * ((s * x) * (c * c))));
	} else {
		tmp = Math.pow((s * (c * x)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = 1.0 / ((s * c) * x)
	tmp = 0
	if c <= -1.4e+154:
		tmp = t_0 * t_0
	elif (c <= -1.7e-160) or not (c <= 5.3e-160):
		tmp = math.cos((x * 2.0)) / (x * (s * ((s * x) * (c * c))))
	else:
		tmp = math.pow((s * (c * x)), -2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(1.0 / Float64(Float64(s * c) * x))
	tmp = 0.0
	if (c <= -1.4e+154)
		tmp = Float64(t_0 * t_0);
	elseif ((c <= -1.7e-160) || !(c <= 5.3e-160))
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(s * Float64(Float64(s * x) * Float64(c * c)))));
	else
		tmp = Float64(s * Float64(c * x)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = 1.0 / ((s * c) * x);
	tmp = 0.0;
	if (c <= -1.4e+154)
		tmp = t_0 * t_0;
	elseif ((c <= -1.7e-160) || ~((c <= 5.3e-160)))
		tmp = cos((x * 2.0)) / (x * (s * ((s * x) * (c * c))));
	else
		tmp = (s * (c * x)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(1.0 / N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.4e+154], N[(t$95$0 * t$95$0), $MachinePrecision], If[Or[LessEqual[c, -1.7e-160], N[Not[LessEqual[c, 5.3e-160]], $MachinePrecision]], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(s * N[(N[(s * x), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(s * N[(c * x), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.4e154

   1. Initial program 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-*l* 37.5%

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

      3. associate-*r* 37.5%

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

      4. *-commutative 37.5%

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

      5. unpow2 37.5%

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

      6. associate-*r* 44.7%

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

      7. associate-*r* 44.7%

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

      8. *-commutative 44.7%

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

      9. unpow2 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in s around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. unpow2 37.5%

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

      4. *-commutative 37.5%

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

      5. swap-sqr 50.8%

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

      6. unpow2 50.8%

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

      7. swap-sqr 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 87.9%

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

      10. *-commutative 87.9%

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

      11. associate-*r* 87.7%

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

      12. unpow2 87.7%

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

      13. associate-*r* 99.5%

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

      14. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 37.5%

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

      1. unpow2 37.5%

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

      2. associate-/r* 37.5%

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

      3. unpow2 37.5%

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

      4. unpow2 37.5%

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

      5. swap-sqr 50.8%

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

      6. unpow2 50.8%

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

      7. associate-/l/ 50.8%

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

      8. *-commutative 50.8%

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

      9. unpow2 50.8%

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

      10. swap-sqr 77.8%

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

      11. unpow2 77.8%

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

      12. /-rgt-identity 77.8%

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

      13. unpow2 77.8%

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

      14. associate-/l* 77.8%

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

      15. associate-/l* 77.8%

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

      16. associate-*l/ 77.8%

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

      17. unpow-1 77.8%

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

      18. unpow-1 77.8%

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

      19. pow-sqr 77.8%

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

      20. metadata-eval 77.8%

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

   9. Simplified 76.3%

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

       1. metadata-eval 87.6%

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

       2. pow-prod-up 87.6%

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

       3. unpow-1 87.6%

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

       4. associate-*r* 87.6%

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

       5. unpow-1 87.6%

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

       6. associate-*r* 99.4%

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

   11. Applied egg-rr 77.8%

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

   if -1.4e154 < c < -1.70000000000000011e-160 or 5.3000000000000001e-160 < c

   1. Initial program 79.5%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-*r* 80.2%

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

      4. unpow2 80.2%

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

      5. unpow2 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in c around 0 80.2%

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

      1. unpow2 80.2%

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

      2. associate-*l* 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in c around 0 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*r* 81.2%

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

      3. unpow2 81.2%

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

      4. associate-*r* 90.7%

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

      5. associate-*l* 94.0%

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

      6. unpow2 94.0%

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

      7. associate-*l* 97.3%

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

      8. *-commutative 97.3%

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

      9. *-commutative 97.3%

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

      10. associate-*r* 94.0%

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

   9. Simplified 94.0%

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

   if -1.70000000000000011e-160 < c < 5.3000000000000001e-160

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*l* 50.1%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. unpow2 50.2%

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

      6. associate-*r* 57.0%

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

      7. associate-*r* 57.0%

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

      8. *-commutative 57.0%

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

      9. unpow2 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in s around 0 50.1%

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

      1. *-commutative 50.1%

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

      2. unpow2 50.1%

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

      3. unpow2 50.1%

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

      4. *-commutative 50.1%

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

      5. swap-sqr 61.4%

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

      6. unpow2 61.4%

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

      7. swap-sqr 93.7%

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

      8. *-commutative 93.7%

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

      9. associate-*r* 92.0%

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

      10. *-commutative 92.0%

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

      11. associate-*r* 97.9%

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

      12. unpow2 97.9%

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

      13. associate-*r* 93.7%

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

      14. *-commutative 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 48.1%

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

      1. unpow2 48.1%

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

      2. associate-/r* 47.9%

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

      3. unpow2 47.9%

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

      4. unpow2 47.9%

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

      5. swap-sqr 55.9%

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

      6. unpow2 55.9%

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

      7. associate-/l/ 56.2%

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

      8. *-commutative 56.2%

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

      9. unpow2 56.2%

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

      10. swap-sqr 69.2%

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

      11. unpow2 69.2%

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

      12. /-rgt-identity 69.2%

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

      13. unpow2 69.2%

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

      14. associate-/l* 69.2%

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

      15. associate-/l* 69.2%

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

      16. associate-*l/ 69.2%

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

      17. unpow-1 69.2%

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

      18. unpow-1 69.2%

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

      19. pow-sqr 69.2%

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

      20. metadata-eval 69.2%

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

   9. Simplified 68.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(s \cdot c\right) \cdot x} \cdot \frac{1}{\left(s \cdot c\right) \cdot x}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-160} \lor \neg \left(c \leq 5.3 \cdot 10^{-160}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\\ \end{array} \]

Alternative 3: 83.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := \frac{1}{\left(s \cdot c\right) \cdot x}\\ \mathbf{if}\;c \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;t_1 \cdot t_1\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-160} \lor \neg \left(c \leq 2.2 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{t_0}{x \cdot \left(s \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (/ 1.0 (* (* s c) x))))
   (if (<= c -1.4e+154)
     (* t_1 t_1)
     (if (or (<= c -9.5e-160) (not (<= c 2.2e-162)))
       (/ t_0 (* x (* s (* (* s x) (* c c)))))
       (/ t_0 (* x (* (* c (* c x)) (* s s))))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = 1.0 / ((s * c) * x);
	double tmp;
	if (c <= -1.4e+154) {
		tmp = t_1 * t_1;
	} else if ((c <= -9.5e-160) || !(c <= 2.2e-162)) {
		tmp = t_0 / (x * (s * ((s * x) * (c * c))));
	} else {
		tmp = t_0 / (x * ((c * (c * x)) * (s * s)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = 1.0d0 / ((s * c) * x)
    if (c <= (-1.4d+154)) then
        tmp = t_1 * t_1
    else if ((c <= (-9.5d-160)) .or. (.not. (c <= 2.2d-162))) then
        tmp = t_0 / (x * (s * ((s * x) * (c * c))))
    else
        tmp = t_0 / (x * ((c * (c * x)) * (s * s)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = 1.0 / ((s * c) * x);
	double tmp;
	if (c <= -1.4e+154) {
		tmp = t_1 * t_1;
	} else if ((c <= -9.5e-160) || !(c <= 2.2e-162)) {
		tmp = t_0 / (x * (s * ((s * x) * (c * c))));
	} else {
		tmp = t_0 / (x * ((c * (c * x)) * (s * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = 1.0 / ((s * c) * x)
	tmp = 0
	if c <= -1.4e+154:
		tmp = t_1 * t_1
	elif (c <= -9.5e-160) or not (c <= 2.2e-162):
		tmp = t_0 / (x * (s * ((s * x) * (c * c))))
	else:
		tmp = t_0 / (x * ((c * (c * x)) * (s * s)))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(1.0 / Float64(Float64(s * c) * x))
	tmp = 0.0
	if (c <= -1.4e+154)
		tmp = Float64(t_1 * t_1);
	elseif ((c <= -9.5e-160) || !(c <= 2.2e-162))
		tmp = Float64(t_0 / Float64(x * Float64(s * Float64(Float64(s * x) * Float64(c * c)))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(c * Float64(c * x)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = 1.0 / ((s * c) * x);
	tmp = 0.0;
	if (c <= -1.4e+154)
		tmp = t_1 * t_1;
	elseif ((c <= -9.5e-160) || ~((c <= 2.2e-162)))
		tmp = t_0 / (x * (s * ((s * x) * (c * c))));
	else
		tmp = t_0 / (x * ((c * (c * x)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.4e+154], N[(t$95$1 * t$95$1), $MachinePrecision], If[Or[LessEqual[c, -9.5e-160], N[Not[LessEqual[c, 2.2e-162]], $MachinePrecision]], N[(t$95$0 / N[(x * N[(s * N[(N[(s * x), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.4e154

   1. Initial program 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-*l* 37.5%

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

      3. associate-*r* 37.5%

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

      4. *-commutative 37.5%

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

      5. unpow2 37.5%

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

      6. associate-*r* 44.7%

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

      7. associate-*r* 44.7%

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

      8. *-commutative 44.7%

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

      9. unpow2 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in s around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. unpow2 37.5%

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

      4. *-commutative 37.5%

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

      5. swap-sqr 50.8%

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

      6. unpow2 50.8%

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

      7. swap-sqr 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 87.9%

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

      10. *-commutative 87.9%

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

      11. associate-*r* 87.7%

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

      12. unpow2 87.7%

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

      13. associate-*r* 99.5%

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

      14. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 37.5%

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

      1. unpow2 37.5%

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

      2. associate-/r* 37.5%

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

      3. unpow2 37.5%

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

      4. unpow2 37.5%

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

      5. swap-sqr 50.8%

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

      6. unpow2 50.8%

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

      7. associate-/l/ 50.8%

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

      8. *-commutative 50.8%

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

      9. unpow2 50.8%

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

      10. swap-sqr 77.8%

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

      11. unpow2 77.8%

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

      12. /-rgt-identity 77.8%

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

      13. unpow2 77.8%

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

      14. associate-/l* 77.8%

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

      15. associate-/l* 77.8%

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

      16. associate-*l/ 77.8%

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

      17. unpow-1 77.8%

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

      18. unpow-1 77.8%

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

      19. pow-sqr 77.8%

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

      20. metadata-eval 77.8%

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

   9. Simplified 76.3%

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

       1. metadata-eval 87.6%

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

       2. pow-prod-up 87.6%

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

       3. unpow-1 87.6%

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

       4. associate-*r* 87.6%

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

       5. unpow-1 87.6%

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

       6. associate-*r* 99.4%

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

   11. Applied egg-rr 77.8%

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

   if -1.4e154 < c < -9.5000000000000002e-160 or 2.1999999999999999e-162 < c

   1. Initial program 79.7%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-*r* 80.1%

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

      4. unpow2 80.1%

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

      5. unpow2 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in c around 0 80.1%

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

      1. unpow2 80.1%

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

      2. associate-*l* 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in c around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*r* 81.2%

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

      3. unpow2 81.2%

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

      4. associate-*r* 90.7%

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

      5. associate-*l* 94.0%

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

      6. unpow2 94.0%

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

      7. associate-*l* 97.3%

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

      8. *-commutative 97.3%

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

      9. *-commutative 97.3%

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

      10. associate-*r* 94.0%

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

   9. Simplified 94.0%

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

   if -9.5000000000000002e-160 < c < 2.1999999999999999e-162

   1. Initial program 54.4%

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

      1. associate-*r* 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-*r* 55.7%

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

      4. unpow2 55.7%

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

      5. unpow2 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in c around 0 55.7%

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

      1. unpow2 55.7%

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

      2. associate-*l* 69.1%

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

   6. Simplified 69.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(s \cdot c\right) \cdot x} \cdot \frac{1}{\left(s \cdot c\right) \cdot x}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-160} \lor \neg \left(c \leq 2.2 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 4: 79.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;{\left(\left(s \cdot c\right) \cdot x\right)}^{-2}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{s}^{-1}}{c \cdot x} \cdot \frac{1}{s \cdot \left(c \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c -1.25e+67)
   (pow (* (* s c) x) -2.0)
   (if (<= c -5.8e-143)
     (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
     (* (/ (pow s -1.0) (* c x)) (/ 1.0 (* s (* c x)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= -1.25e+67) {
		tmp = pow(((s * c) * x), -2.0);
	} else if (c <= -5.8e-143) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = (pow(s, -1.0) / (c * x)) * (1.0 / (s * (c * x)));
	}
	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 (c <= (-1.25d+67)) then
        tmp = ((s * c) * x) ** (-2.0d0)
    else if (c <= (-5.8d-143)) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = ((s ** (-1.0d0)) / (c * x)) * (1.0d0 / (s * (c * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= -1.25e+67) {
		tmp = Math.pow(((s * c) * x), -2.0);
	} else if (c <= -5.8e-143) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = (Math.pow(s, -1.0) / (c * x)) * (1.0 / (s * (c * x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= -1.25e+67:
		tmp = math.pow(((s * c) * x), -2.0)
	elif c <= -5.8e-143:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = (math.pow(s, -1.0) / (c * x)) * (1.0 / (s * (c * x)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= -1.25e+67)
		tmp = Float64(Float64(s * c) * x) ^ -2.0;
	elseif (c <= -5.8e-143)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = Float64(Float64((s ^ -1.0) / Float64(c * x)) * Float64(1.0 / Float64(s * Float64(c * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= -1.25e+67)
		tmp = ((s * c) * x) ^ -2.0;
	elseif (c <= -5.8e-143)
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = ((s ^ -1.0) / (c * x)) * (1.0 / (s * (c * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, -1.25e+67], N[Power[N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[c, -5.8e-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[(N[(N[Power[s, -1.0], $MachinePrecision] / N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.24999999999999994e67

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*l* 60.2%

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

      3. associate-*r* 57.5%

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

      4. *-commutative 57.5%

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

      5. unpow2 57.5%

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

      6. associate-*r* 63.4%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. unpow2 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in s around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. unpow2 60.2%

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

      3. unpow2 60.2%

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

      4. *-commutative 60.2%

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

      5. swap-sqr 74.7%

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

      6. unpow2 74.7%

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

      7. swap-sqr 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r* 94.3%

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

      10. *-commutative 94.3%

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

      11. associate-*r* 94.3%

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

      12. unpow2 94.3%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 57.9%

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

      1. unpow2 57.9%

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

      2. associate-/r* 57.9%

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

      3. unpow2 57.9%

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

      4. unpow2 57.9%

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

      5. swap-sqr 72.4%

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

      6. unpow2 72.4%

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

      7. associate-/l/ 72.4%

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

      8. *-commutative 72.4%

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

      9. unpow2 72.4%

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

      10. swap-sqr 86.6%

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

      11. unpow2 86.6%

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

      12. /-rgt-identity 86.6%

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

      13. unpow2 86.6%

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

      14. associate-/l* 86.6%

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

      15. associate-/l* 86.6%

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

      16. associate-*l/ 86.5%

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

      17. unpow-1 86.5%

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

      18. unpow-1 86.5%

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

      19. pow-sqr 86.6%

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

      20. metadata-eval 86.6%

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

   9. Simplified 85.9%

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

   10. Taylor expanded in s around 0 86.6%

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

       1. associate-*r* 86.7%

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

       2. *-commutative 86.7%

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

       3. *-commutative 86.7%

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

   12. Simplified 86.7%

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

   if -1.24999999999999994e67 < c < -5.8000000000000002e-143

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*l* 82.2%

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

      3. associate-*r* 82.1%

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

      4. *-commutative 82.1%

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

      5. unpow2 82.1%

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

      6. associate-*r* 87.0%

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

      7. associate-*r* 87.3%

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

      8. *-commutative 87.3%

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

      9. unpow2 87.3%

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

   3. Simplified 87.3%

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

   if -5.8000000000000002e-143 < c

   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. *-commutative 68.2%

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

      2. associate-*l* 64.7%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r* 70.2%

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

      7. associate-*r* 73.5%

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

      8. *-commutative 73.5%

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

      9. unpow2 73.5%

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

   3. Simplified 73.5%

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

      1. add-cube-cbrt 73.5%

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

      2. times-frac 73.4%

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

      3. associate-*r* 72.6%

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

      4. swap-sqr 89.2%

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

      5. associate-*r* 94.4%

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

      6. *-commutative 94.4%

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

      7. times-frac 94.8%

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

      8. associate-*l* 97.3%

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

      9. add-cube-cbrt 97.5%

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

      10. associate-/r* 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in x around 0 79.6%

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

      1. associate-/r* 79.7%

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

      2. unpow-1 79.7%

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

      3. *-commutative 79.7%

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

   8. Simplified 79.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;{\left(\left(s \cdot c\right) \cdot x\right)}^{-2}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{s}^{-1}}{c \cdot x} \cdot \frac{1}{s \cdot \left(c \cdot x\right)}\\ \end{array} \]

Alternative 5: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in s around 0 66.8%

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

   1. *-commutative 66.8%

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

   2. unpow2 66.8%

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

   3. unpow2 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 81.8%

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

   6. unpow2 81.8%

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

   7. swap-sqr 98.2%

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

   8. *-commutative 98.2%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

   11. associate-*r* 97.3%

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

   12. unpow2 97.3%

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

   13. associate-*r* 98.2%

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

   14. *-commutative 98.2%

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

6. Simplified 98.2%

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

   1. unpow2 98.2%

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

8. Applied egg-rr 98.2%

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

9. Final simplification 98.2%

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

Alternative 6: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*r* 66.8%

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

   3. associate-*r* 66.1%

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

   4. unpow2 66.1%

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

   5. unswap-sqr 80.4%

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

   6. unpow2 80.4%

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

   7. swap-sqr 97.3%

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

   8. *-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. *-commutative 97.3%

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

   11. *-commutative 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 7: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

   1. add-cube-cbrt 74.5%

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

   2. times-frac 74.5%

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

   3. associate-*r* 73.2%

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

   4. swap-sqr 88.4%

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

   5. associate-*r* 94.3%

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

   6. *-commutative 94.3%

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

   7. times-frac 94.6%

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

   8. associate-*l* 97.1%

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

   9. add-cube-cbrt 97.3%

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

   10. associate-/r* 97.4%

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

5. Applied egg-rr 97.3%

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

   1. un-div-inv 97.4%

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

   2. associate-*r* 96.0%

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

   3. associate-*r* 98.2%

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

7. Applied egg-rr 98.2%

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

8. Final simplification 98.2%

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

Alternative 8: 79.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in s around 0 66.8%

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

   1. *-commutative 66.8%

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

   2. unpow2 66.8%

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

   3. unpow2 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 81.8%

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

   6. unpow2 81.8%

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

   7. swap-sqr 98.2%

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

   8. *-commutative 98.2%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

   11. associate-*r* 97.3%

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

   12. unpow2 97.3%

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

   13. associate-*r* 98.2%

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

   14. *-commutative 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in x around 0 61.2%

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

   1. unpow2 61.2%

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

   2. associate-/r* 60.7%

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

   3. unpow2 60.7%

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

   4. unpow2 60.7%

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

   5. swap-sqr 70.9%

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

   6. unpow2 70.9%

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

   7. associate-/l/ 71.4%

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

   8. *-commutative 71.4%

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

   9. unpow2 71.4%

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

   10. swap-sqr 79.6%

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

   11. unpow2 79.6%

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

   12. /-rgt-identity 79.6%

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

   13. unpow2 79.6%

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

   14. associate-/l* 79.6%

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

   15. associate-/l* 79.6%

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

   16. associate-*l/ 79.6%

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

   17. unpow-1 79.6%

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

   18. unpow-1 79.6%

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

   19. pow-sqr 79.6%

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

   20. metadata-eval 79.6%

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

9. Simplified 79.0%

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

10. Taylor expanded in s around 0 79.6%

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

    1. associate-*r* 79.7%

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

    2. *-commutative 79.7%

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

    3. *-commutative 79.7%

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

12. Simplified 79.7%

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

13. Final simplification 79.7%

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

Alternative 9: 56.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{s \cdot s}}{c \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 4.6e+152)
   (/ 1.0 (* (* c c) (* (* s s) (* x x))))
   (/ (/ -2.0 (* s s)) (* c c))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.6e+152) {
		tmp = 1.0 / ((c * c) * ((s * s) * (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 <= 4.6d+152) then
        tmp = 1.0d0 / ((c * c) * ((s * s) * (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 <= 4.6e+152) {
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
	} else {
		tmp = (-2.0 / (s * s)) / (c * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 4.6e+152:
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)))
	else:
		tmp = (-2.0 / (s * s)) / (c * c)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 4.6e+152)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(s * s) * Float64(x * x))));
	else
		tmp = Float64(Float64(-2.0 / Float64(s * s)) / Float64(c * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 4.6e+152)
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
	else
		tmp = (-2.0 / (s * s)) / (c * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 4.6e+152], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 4.5999999999999997e152

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*l* 68.0%

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

      3. associate-*r* 67.4%

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

      4. *-commutative 67.4%

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

      5. unpow2 67.4%

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

      6. associate-*r* 72.9%

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

      7. associate-*r* 75.6%

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

      8. *-commutative 75.6%

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

      9. unpow2 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in s around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. unpow2 67.9%

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

      3. unpow2 68.0%

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

      4. *-commutative 68.0%

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

      5. swap-sqr 80.4%

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

      6. unpow2 80.4%

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

      7. swap-sqr 97.9%

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

      8. *-commutative 97.9%

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

      9. associate-*r* 95.4%

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

      10. *-commutative 95.4%

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

      11. associate-*r* 97.0%

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

      12. unpow2 97.0%

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

      13. associate-*r* 97.9%

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

      14. *-commutative 97.9%

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

   6. Simplified 97.9%

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

      1. unpow2 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in x around inf 67.2%

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

       1. *-commutative 67.2%

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

       2. associate-*r* 67.4%

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

       3. unpow2 67.4%

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

       4. unpow2 67.4%

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

       5. unswap-sqr 80.9%

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

       6. unpow2 80.9%

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

       7. swap-sqr 98.3%

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

       8. associate-/r* 98.3%

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

       9. *-commutative 98.3%

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

       10. associate-*r* 96.8%

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

       11. *-lft-identity 96.8%

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

       12. times-frac 94.9%

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

       13. associate-/l/ 94.8%

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

       14. times-frac 94.2%

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

       15. associate-*l/ 98.0%

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

       16. associate-/r* 97.9%

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

   11. Simplified 97.0%

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

   12. Taylor expanded in x around 0 60.7%

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

       1. unpow2 60.7%

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

       2. associate-*r* 61.0%

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

       3. *-commutative 61.0%

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

       4. associate-*r* 61.5%

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

       5. unpow2 61.5%

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

       6. unpow2 61.5%

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

       7. unpow2 61.5%

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

       8. *-commutative 61.5%

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

       9. unpow2 61.5%

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

   14. Simplified 61.5%

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

   if 4.5999999999999997e152 < s

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*l* 59.1%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. unpow2 59.1%

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

      6. associate-*r* 65.2%

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

      7. associate-*r* 68.2%

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

      8. *-commutative 68.2%

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

      9. unpow2 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 59.1%

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

      2. unpow2 59.1%

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

      3. unpow2 59.1%

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

      4. unpow2 59.1%

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

      5. *-commutative 59.1%

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

      6. associate-*r/ 59.1%

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

      7. metadata-eval 59.1%

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

      8. unpow2 59.1%

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

      9. *-commutative 59.1%

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

      10. unpow2 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in x around inf 74.0%

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

      1. unpow2 74.0%

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

      2. associate-/r* 74.0%

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

      3. unpow2 74.0%

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

   9. Simplified 74.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{s \cdot s}}{c \cdot c}\\ \end{array} \]

Alternative 10: 79.5% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. associate-/r* 70.5%

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

   2. unpow2 70.5%

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

   3. *-commutative 70.5%

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

   4. unpow2 70.5%

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

3. Simplified 70.5%

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

4. Taylor expanded in x around 0 63.7%

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

   1. unpow2 63.7%

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

6. Simplified 63.7%

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

   1. associate-/l/ 64.2%

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

   2. *-commutative 64.2%

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

   3. associate-*r* 70.2%

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

   4. associate-*r* 71.4%

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

   5. *-commutative 71.4%

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

   6. pow2 71.4%

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

   7. pow2 71.4%

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

   8. unpow-prod-down 79.6%

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

   9. *-commutative 79.6%

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

   10. add-sqr-sqrt 79.5%

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

   11. sqrt-div 79.6%

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

   12. metadata-eval 79.6%

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

   13. unpow2 79.6%

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

   14. sqrt-prod 51.3%

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

   15. add-sqr-sqrt 57.0%

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

   16. sqrt-div 57.0%

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

   17. metadata-eval 57.0%

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

8. Applied egg-rr 79.6%

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

9. Final simplification 79.6%

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

Alternative 11: 79.6% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

   1. add-cube-cbrt 74.5%

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

   2. times-frac 74.5%

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

   3. associate-*r* 73.2%

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

   4. swap-sqr 88.4%

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

   5. associate-*r* 94.3%

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

   6. *-commutative 94.3%

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

   7. times-frac 94.6%

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

   8. associate-*l* 97.1%

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

   9. add-cube-cbrt 97.3%

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

   10. associate-/r* 97.4%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 79.0%

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

   1. *-commutative 79.0%

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

8. Simplified 79.0%

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

9. Final simplification 79.0%

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

Alternative 12: 79.2% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in s around 0 66.8%

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

   1. *-commutative 66.8%

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

   2. unpow2 66.8%

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

   3. unpow2 66.8%

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

   4. *-commutative 66.8%

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

   5. swap-sqr 81.8%

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

   6. unpow2 81.8%

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

   7. swap-sqr 98.2%

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

   8. *-commutative 98.2%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

   11. associate-*r* 97.3%

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

   12. unpow2 97.3%

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

   13. associate-*r* 98.2%

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

   14. *-commutative 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in x around 0 61.2%

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

   1. unpow2 61.2%

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

   2. associate-/r* 60.7%

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

   3. unpow2 60.7%

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

   4. unpow2 60.7%

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

   5. swap-sqr 70.9%

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

   6. unpow2 70.9%

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

   7. associate-/l/ 71.4%

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

   8. *-commutative 71.4%

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

   9. unpow2 71.4%

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

   10. swap-sqr 79.6%

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

   11. unpow2 79.6%

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

   12. /-rgt-identity 79.6%

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

   13. unpow2 79.6%

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

   14. associate-/l* 79.6%

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

   15. associate-/l* 79.6%

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

   16. associate-*l/ 79.6%

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

   17. unpow-1 79.6%

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

   18. unpow-1 79.6%

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

   19. pow-sqr 79.6%

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

   20. metadata-eval 79.6%

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

9. Simplified 79.0%

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

    1. metadata-eval 97.3%

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

    2. pow-prod-up 97.3%

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

    3. unpow-1 97.3%

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

    4. associate-*r* 96.3%

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

    5. unpow-1 96.3%

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

    6. associate-*r* 98.5%

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

11. Applied egg-rr 79.7%

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

12. Final simplification 79.7%

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

Alternative 13: 65.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. associate-/r* 70.5%

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

   2. unpow2 70.5%

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

   3. *-commutative 70.5%

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

   4. unpow2 70.5%

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

3. Simplified 70.5%

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

4. Taylor expanded in x around 0 63.7%

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

   1. unpow2 63.7%

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

6. Simplified 63.7%

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

7. Taylor expanded in x around 0 63.7%

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

   1. unpow2 63.7%

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

   2. associate-*l* 69.7%

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

9. Simplified 69.7%

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

10. Final simplification 69.7%

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

Alternative 14: 28.6% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-2}{s \cdot s}}{c \cdot c} \end{array} \]

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(Float64(-2.0 / 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[(N[(-2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 66.8%

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

   3. associate-*r* 66.3%

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

   4. *-commutative 66.3%

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

   5. unpow2 66.3%

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

   6. associate-*r* 71.8%

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

   7. associate-*r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in x around 0 34.5%

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

   1. associate-/r* 34.2%

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

   2. unpow2 34.2%

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

   3. unpow2 34.2%

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

   4. unpow2 34.2%

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

   5. *-commutative 34.2%

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

   6. associate-*r/ 34.2%

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

   7. metadata-eval 34.2%

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

   8. unpow2 34.2%

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

   9. *-commutative 34.2%

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

   10. unpow2 34.2%

       \[\leadsto \frac{\frac{1}{c \cdot c}}{\left(x \cdot x\right) \cdot \left(s \cdot s\right)} - \frac{2}{\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot s\right)} \]

6. Simplified 34.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot c}}{\left(x \cdot x\right) \cdot \left(s \cdot s\right)} - \frac{2}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}} \]

7. Taylor expanded in x around inf 26.9%

   \[\leadsto \color{blue}{\frac{-2}{{s}^{2} \cdot {c}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 26.9%

      \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}} \]

   2. associate-/r* 26.9%

      \[\leadsto \color{blue}{\frac{\frac{-2}{s \cdot s}}{{c}^{2}}} \]

   3. unpow2 26.9%

      \[\leadsto \frac{\frac{-2}{s \cdot s}}{\color{blue}{c \cdot c}} \]

9. Simplified 26.9%

   \[\leadsto \color{blue}{\frac{\frac{-2}{s \cdot s}}{c \cdot c}} \]

10. Final simplification 26.9%

    \[\leadsto \frac{\frac{-2}{s \cdot s}}{c \cdot c} \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))