mixedcos

Percentage Accurate: 67.4% → 97.3%

Time: 12.4s

Alternatives: 10

Speedup: 24.1×

• 32.6% 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: 67.4% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

   1. div-inv 97.6%

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

   2. *-commutative 97.6%

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

   3. pow2 97.6%

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

   4. pow-flip 97.8%

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

   5. metadata-eval 97.8%

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

5. Applied egg-rr 97.8%

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

6. Final simplification 97.8%

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

Alternative 2: 86.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (cos (* x 2.0)) (* s (* (* x x) (* c (* s c)))))))
   (if (<= x -1.9e-18)
     t_0
     (if (<= x 1.05e-10)
       (/ 1.0 (pow (* c (* x s)) 2.0))
       (if (<= x 5.8e+174) t_0 (/ 1.0 (pow (* s (* x c)) 2.0)))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0)) / (s * ((x * x) * (c * (s * c))));
	double tmp;
	if (x <= -1.9e-18) {
		tmp = t_0;
	} else if (x <= 1.05e-10) {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	} else if (x <= 5.8e+174) {
		tmp = t_0;
	} else {
		tmp = 1.0 / pow((s * (x * c)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0)) / (s * ((x * x) * (c * (s * c))))
    if (x <= (-1.9d-18)) then
        tmp = t_0
    else if (x <= 1.05d-10) then
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    else if (x <= 5.8d+174) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((s * (x * c)) ** 2.0d0)
    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)) / (s * ((x * x) * (c * (s * c))));
	double tmp;
	if (x <= -1.9e-18) {
		tmp = t_0;
	} else if (x <= 1.05e-10) {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	} else if (x <= 5.8e+174) {
		tmp = t_0;
	} else {
		tmp = 1.0 / Math.pow((s * (x * c)), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0)) / (s * ((x * x) * (c * (s * c))))
	tmp = 0
	if x <= -1.9e-18:
		tmp = t_0
	elif x <= 1.05e-10:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	elif x <= 5.8e+174:
		tmp = t_0
	else:
		tmp = 1.0 / math.pow((s * (x * c)), 2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(s * c)))))
	tmp = 0.0
	if (x <= -1.9e-18)
		tmp = t_0;
	elseif (x <= 1.05e-10)
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	elseif (x <= 5.8e+174)
		tmp = t_0;
	else
		tmp = Float64(1.0 / (Float64(s * Float64(x * c)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0)) / (s * ((x * x) * (c * (s * c))));
	tmp = 0.0;
	if (x <= -1.9e-18)
		tmp = t_0;
	elseif (x <= 1.05e-10)
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	elseif (x <= 5.8e+174)
		tmp = t_0;
	else
		tmp = 1.0 / ((s * (x * c)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e-18], t$95$0, If[LessEqual[x, 1.05e-10], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+174], t$95$0, N[(1.0 / N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999e-18 or 1.05e-10 < x < 5.7999999999999999e174

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*l* 60.6%

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

      3. associate-*r* 60.7%

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

      4. *-commutative 60.7%

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

      5. unpow2 60.7%

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

      6. associate-*r* 69.2%

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

      7. associate-*r* 71.1%

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

      8. *-commutative 71.1%

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

      9. unpow2 71.1%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in c around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. associate-*l* 78.5%

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

   6. Simplified 78.5%

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

   if -1.8999999999999999e-18 < x < 1.05e-10

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 65.4%

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

      3. associate-*r* 67.0%

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

      4. unpow2 67.0%

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

      5. unswap-sqr 81.4%

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

      6. unpow2 81.4%

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

      7. swap-sqr 97.2%

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

      8. *-commutative 97.2%

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

      9. *-commutative 97.2%

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

      10. *-commutative 97.2%

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

      11. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. associate-*r* 64.8%

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

      4. *-commutative 64.8%

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

      5. associate-*r* 67.0%

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

      6. unpow2 67.0%

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

      7. unpow2 67.0%

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

      8. unpow2 67.0%

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

      9. associate-/r* 67.0%

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

      10. associate-/r* 67.0%

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

      11. unpow2 67.0%

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

      12. *-commutative 67.0%

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

      13. unpow2 67.0%

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

      14. swap-sqr 81.4%

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

      15. unpow2 81.4%

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

      16. associate-/l/ 89.2%

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

      17. associate-/r* 89.2%

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

      18. *-commutative 89.2%

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

      19. associate-*r* 81.4%

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

   6. Simplified 98.1%

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

   if 5.7999999999999999e174 < x

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*r* 40.3%

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

      3. associate-*r* 40.3%

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

      4. unpow2 40.3%

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

      5. unswap-sqr 74.1%

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

      6. unpow2 74.1%

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

      7. swap-sqr 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. Taylor expanded in x around 0 40.3%

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

      1. unpow2 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

   6. Simplified 40.3%

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

      1. *-commutative 40.3%

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

      2. unswap-sqr 57.6%

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

      3. *-commutative 57.6%

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

      4. *-commutative 57.6%

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

      5. unpow2 57.6%

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

      6. pow2 57.6%

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

      7. pow-prod-down 67.4%

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

      8. *-commutative 67.4%

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

      9. associate-*r* 67.4%

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

   8. Applied egg-rr 67.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \]

Alternative 3: 87.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := \frac{t_0}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (/ t_0 (* s (* (* x x) (* c (* s c)))))))
   (if (<= x -2.7e-12)
     t_1
     (if (<= x 9.5e-11)
       (/ 1.0 (pow (* c (* x s)) 2.0))
       (if (<= x 2e+150) t_1 (/ t_0 (* x (* (* c c) (* s (* x s))))))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = t_0 / (s * ((x * x) * (c * (s * c))));
	double tmp;
	if (x <= -2.7e-12) {
		tmp = t_1;
	} else if (x <= 9.5e-11) {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	} else if (x <= 2e+150) {
		tmp = t_1;
	} else {
		tmp = t_0 / (x * ((c * c) * (s * (x * 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 = t_0 / (s * ((x * x) * (c * (s * c))))
    if (x <= (-2.7d-12)) then
        tmp = t_1
    else if (x <= 9.5d-11) then
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    else if (x <= 2d+150) then
        tmp = t_1
    else
        tmp = t_0 / (x * ((c * c) * (s * (x * 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 = t_0 / (s * ((x * x) * (c * (s * c))));
	double tmp;
	if (x <= -2.7e-12) {
		tmp = t_1;
	} else if (x <= 9.5e-11) {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	} else if (x <= 2e+150) {
		tmp = t_1;
	} else {
		tmp = t_0 / (x * ((c * c) * (s * (x * s))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = t_0 / (s * ((x * x) * (c * (s * c))))
	tmp = 0
	if x <= -2.7e-12:
		tmp = t_1
	elif x <= 9.5e-11:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	elif x <= 2e+150:
		tmp = t_1
	else:
		tmp = t_0 / (x * ((c * c) * (s * (x * s))))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(t_0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(s * c)))))
	tmp = 0.0
	if (x <= -2.7e-12)
		tmp = t_1;
	elseif (x <= 9.5e-11)
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	elseif (x <= 2e+150)
		tmp = t_1;
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(c * c) * Float64(s * Float64(x * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = t_0 / (s * ((x * x) * (c * (s * c))));
	tmp = 0.0;
	if (x <= -2.7e-12)
		tmp = t_1;
	elseif (x <= 9.5e-11)
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	elseif (x <= 2e+150)
		tmp = t_1;
	else
		tmp = t_0 / (x * ((c * c) * (s * (x * 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[(t$95$0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-12], t$95$1, If[LessEqual[x, 9.5e-11], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+150], t$95$1, N[(t$95$0 / N[(x * N[(N[(c * c), $MachinePrecision] * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6999999999999998e-12 or 9.49999999999999951e-11 < x < 1.99999999999999996e150

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l* 61.2%

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

      3. associate-*r* 61.3%

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

      4. *-commutative 61.3%

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

      5. unpow2 61.3%

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

      6. associate-*r* 69.9%

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

      7. associate-*r* 71.7%

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

      8. *-commutative 71.7%

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

      9. unpow2 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in c around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. unpow2 71.7%

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

      3. associate-*l* 79.2%

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

   6. Simplified 79.2%

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

   if -2.6999999999999998e-12 < x < 9.49999999999999951e-11

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 65.4%

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

      3. associate-*r* 67.0%

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

      4. unpow2 67.0%

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

      5. unswap-sqr 81.4%

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

      6. unpow2 81.4%

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

      7. swap-sqr 97.2%

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

      8. *-commutative 97.2%

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

      9. *-commutative 97.2%

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

      10. *-commutative 97.2%

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

      11. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. associate-*r* 64.8%

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

      4. *-commutative 64.8%

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

      5. associate-*r* 67.0%

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

      6. unpow2 67.0%

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

      7. unpow2 67.0%

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

      8. unpow2 67.0%

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

      9. associate-/r* 67.0%

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

      10. associate-/r* 67.0%

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

      11. unpow2 67.0%

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

      12. *-commutative 67.0%

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

      13. unpow2 67.0%

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

      14. swap-sqr 81.4%

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

      15. unpow2 81.4%

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

      16. associate-/l/ 89.2%

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

      17. associate-/r* 89.2%

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

      18. *-commutative 89.2%

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

      19. associate-*r* 81.4%

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

   6. Simplified 98.1%

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

   if 1.99999999999999996e150 < x

   1. Initial program 52.9%

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

      1. associate-*r* 60.1%

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

      2. *-commutative 60.1%

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

      3. *-commutative 60.1%

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

      4. associate-*r* 54.7%

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

      5. *-commutative 54.7%

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

      6. unpow2 54.7%

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

      7. unpow2 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in x around 0 59.7%

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

      1. unpow2 59.7%

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

      2. *-commutative 59.7%

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

      3. associate-*r* 60.1%

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

      4. unpow2 60.1%

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

      5. associate-*r* 79.7%

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

      6. *-commutative 79.7%

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

   6. Simplified 79.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 4: 81.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c -1.15e+84)
   (/ 1.0 (pow (* s (* x c)) 2.0))
   (if (<= c -2.75e-157)
     (/ (cos (* x 2.0)) (* (* c c) (* x (* s (* x s)))))
     (/ 1.0 (pow (* c (* x s)) 2.0)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= -1.15e+84) {
		tmp = 1.0 / pow((s * (x * c)), 2.0);
	} else if (c <= -2.75e-157) {
		tmp = cos((x * 2.0)) / ((c * c) * (x * (s * (x * s))));
	} else {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c <= (-1.15d+84)) then
        tmp = 1.0d0 / ((s * (x * c)) ** 2.0d0)
    else if (c <= (-2.75d-157)) then
        tmp = cos((x * 2.0d0)) / ((c * c) * (x * (s * (x * s))))
    else
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= -1.15e+84) {
		tmp = 1.0 / Math.pow((s * (x * c)), 2.0);
	} else if (c <= -2.75e-157) {
		tmp = Math.cos((x * 2.0)) / ((c * c) * (x * (s * (x * s))));
	} else {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= -1.15e+84:
		tmp = 1.0 / math.pow((s * (x * c)), 2.0)
	elif c <= -2.75e-157:
		tmp = math.cos((x * 2.0)) / ((c * c) * (x * (s * (x * s))))
	else:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= -1.15e+84)
		tmp = Float64(1.0 / (Float64(s * Float64(x * c)) ^ 2.0));
	elseif (c <= -2.75e-157)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(Float64(c * c) * Float64(x * Float64(s * Float64(x * s)))));
	else
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= -1.15e+84)
		tmp = 1.0 / ((s * (x * c)) ^ 2.0);
	elseif (c <= -2.75e-157)
		tmp = cos((x * 2.0)) / ((c * c) * (x * (s * (x * s))));
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, -1.15e+84], N[(1.0 / N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.75e-157], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(c * c), $MachinePrecision] * N[(x * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1499999999999999e84

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*r* 55.9%

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

      3. associate-*r* 60.2%

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

      4. unpow2 60.2%

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

      5. unswap-sqr 78.5%

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

      6. unpow2 78.5%

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

      7. swap-sqr 96.4%

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

      8. *-commutative 96.4%

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

      9. *-commutative 96.4%

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

      10. *-commutative 96.4%

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

      11. *-commutative 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in x around 0 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. unpow2 55.9%

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

   6. Simplified 55.9%

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

      1. *-commutative 55.9%

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

      2. unswap-sqr 67.7%

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

      3. *-commutative 67.7%

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

      4. *-commutative 67.7%

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

      5. unpow2 67.7%

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

      6. pow2 67.7%

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

      7. pow-prod-down 93.1%

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

      8. *-commutative 93.1%

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

      9. associate-*r* 94.2%

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

   8. Applied egg-rr 94.2%

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

   if -1.1499999999999999e84 < c < -2.7499999999999999e-157

   1. Initial program 75.3%

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

      1. unpow2 75.3%

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

      2. *-commutative 75.3%

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

      3. unpow2 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in x around 0 75.3%

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

      1. unpow2 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

   6. Simplified 89.5%

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

   if -2.7499999999999999e-157 < c

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*r* 58.7%

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

      3. associate-*r* 58.6%

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

      4. unpow2 58.6%

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

      5. unswap-sqr 76.1%

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

      6. unpow2 76.1%

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

      7. swap-sqr 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. *-commutative 98.4%

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

      11. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around 0 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. associate-*r* 51.5%

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

      4. *-commutative 51.5%

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

      5. associate-*r* 53.1%

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

      6. unpow2 53.1%

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

      7. unpow2 53.1%

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

      8. unpow2 53.1%

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

      9. associate-/r* 53.0%

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

      10. associate-/r* 53.0%

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

      11. unpow2 53.0%

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

      12. *-commutative 53.0%

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

      13. unpow2 53.0%

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

      14. swap-sqr 62.8%

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

      15. unpow2 62.8%

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

      16. associate-/l/ 70.6%

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

      17. associate-/r* 70.6%

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

      18. *-commutative 70.6%

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

      19. associate-*r* 62.8%

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

   6. Simplified 76.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

   1. div-inv 97.6%

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

   2. *-commutative 97.6%

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

   3. pow2 97.6%

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

   4. pow-flip 97.8%

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

   5. metadata-eval 97.8%

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

5. Applied egg-rr 97.8%

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

   1. metadata-eval 97.8%

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

   2. pow-prod-up 97.7%

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

   3. unpow-1 97.7%

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

   4. unpow-1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. un-div-inv 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

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

Alternative 6: 94.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in s around 0 95.5%

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

5. Final simplification 95.5%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = s * (x * c)
    code = cos((x * 2.0d0)) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

4. Final simplification 97.5%

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

Alternative 8: 77.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around 0 54.9%

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

   1. unpow2 54.9%

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

   2. unpow2 54.9%

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

   3. associate-*r* 54.6%

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

   4. *-commutative 54.6%

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

   5. associate-*r* 55.7%

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

   6. unpow2 55.7%

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

   7. unpow2 55.7%

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

   8. unpow2 55.7%

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

   9. associate-/r* 55.7%

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

   10. associate-/r* 55.7%

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

   11. unpow2 55.7%

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

   12. *-commutative 55.7%

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

   13. unpow2 55.7%

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

   14. swap-sqr 65.7%

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

   15. unpow2 65.7%

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

   16. associate-/l/ 73.8%

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

   17. associate-/r* 73.7%

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

   18. *-commutative 73.7%

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

   19. associate-*r* 65.7%

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

6. Simplified 78.4%

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

7. Final simplification 78.4%

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

Alternative 9: 77.7% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. associate-/r* 64.3%

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

   2. unpow2 64.3%

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

   3. *-commutative 64.3%

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

   4. unpow2 64.3%

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

3. Simplified 64.3%

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

4. Taylor expanded in x around 0 57.2%

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

   1. unpow2 57.2%

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

6. Simplified 57.2%

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

   1. div-inv 57.2%

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

   2. inv-pow 57.2%

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

   3. inv-pow 57.2%

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

   4. associate-*r* 54.9%

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

   5. pow2 54.9%

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

   6. pow2 54.9%

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

   7. unpow-prod-down 66.0%

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

   8. *-commutative 66.0%

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

   9. unpow-prod-down 66.1%

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

   10. associate-*r* 73.5%

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

   11. inv-pow 73.5%

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

   12. add-sqr-sqrt 73.4%

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

8. Applied egg-rr 78.3%

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

9. Final simplification 78.3%

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

Alternative 10: 72.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 59.8%

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

   3. associate-*r* 60.5%

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

   4. unpow2 60.5%

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

   5. unswap-sqr 76.7%

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

   6. unpow2 76.7%

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

   7. swap-sqr 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

   10. *-commutative 97.5%

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

   11. *-commutative 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around 0 54.9%

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

   1. unpow2 54.9%

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

   2. unpow2 54.9%

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

   3. unpow2 54.9%

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

6. Simplified 54.9%

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

7. Taylor expanded in c around 0 55.7%

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

   1. unpow2 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. associate-*r* 54.6%

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

   5. *-commutative 54.6%

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

   6. associate-*r* 54.9%

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

   7. swap-sqr 66.1%

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

   8. unpow2 66.1%

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

   9. associate-*r* 73.5%

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

9. Simplified 73.5%

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

    1. unpow2 73.5%

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

11. Applied egg-rr 73.5%

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

12. Final simplification 73.5%

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

Reproduce

herbie shell --seed 2023194 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))