mixedcos

Percentage Accurate: 67.0% → 98.8%

Time: 19.9s

Alternatives: 19

Speedup: 24.1×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = t_0 / pow((c * (x * s)), 2.0);
	} else {
		tmp = t_0 / pow((x * (c * s)), 2.0);
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x))) < +inf.0

   1. Initial program 85.1%

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

      1. associate-/r* 83.6%

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

      2. remove-double-neg 83.6%

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

      3. distribute-lft-neg-out 83.6%

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

      4. distribute-lft-neg-out 83.6%

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

      5. distribute-rgt-neg-out 83.6%

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

      6. associate-/r* 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*r* 84.1%

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

      9. associate-*r* 85.1%

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

      10. associate-*r* 78.2%

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

      11. sqr-neg 78.2%

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

      12. associate-*r* 85.1%

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

      13. *-commutative 85.1%

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

      14. unpow2 85.1%

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

      15. sqr-neg 85.1%

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

      16. unpow2 85.1%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in x around inf 78.2%

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

      1. count-2 78.2%

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

      2. *-lft-identity 78.2%

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

      3. associate-*r* 78.2%

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

      4. unpow2 78.2%

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

      5. unpow2 78.2%

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

      6. associate-*r* 80.0%

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

      7. unpow2 80.0%

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

      8. *-commutative 80.0%

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

      9. times-frac 79.3%

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

      10. associate-*r* 77.5%

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

      11. swap-sqr 82.1%

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

      12. unpow2 82.1%

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

      13. associate-/r* 82.1%

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

      14. times-frac 90.1%

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

      15. *-commutative 90.1%

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

      16. associate-/l* 89.8%

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

   6. Simplified 99.5%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. remove-double-neg 0.0%

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

      3. distribute-lft-neg-out 0.0%

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

      4. distribute-lft-neg-out 0.0%

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

      5. distribute-rgt-neg-out 0.0%

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

      6. associate-/r* 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-*r* 0.4%

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

      9. associate-*r* 0.0%

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

      10. associate-*r* 0.0%

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

      11. sqr-neg 0.0%

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

      12. associate-*r* 0.0%

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

      13. *-commutative 0.0%

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

      14. unpow2 0.0%

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

      15. sqr-neg 0.0%

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

      16. unpow2 0.0%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in x around inf 0.0%

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

      1. count-2 0.0%

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

      2. *-lft-identity 0.0%

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

      3. associate-*r* 0.2%

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

      4. unpow2 0.2%

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

      5. unpow2 0.2%

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

      6. associate-*r* 25.8%

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

      7. unpow2 25.8%

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

      8. *-commutative 25.8%

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

      9. times-frac 25.6%

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

      10. associate-*r* 0.1%

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

      11. swap-sqr 72.6%

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

      12. unpow2 72.6%

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

      13. associate-/r* 72.6%

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

      14. times-frac 85.0%

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

      15. *-commutative 85.0%

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

      16. associate-/l* 85.0%

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

   6. Simplified 76.7%

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

      1. add-sqr-sqrt_binary64 43.9%

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

   8. Applied rewrite-once 43.9%

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

      1. rem-square-sqrt 76.7%

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

      2. associate-*r* 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 99.4%

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

Alternative 2: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 13000000000:\\ \;\;\;\;\frac{t_0}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 13000000000.0)
     (/ t_0 (pow (* c (* x s)) 2.0))
     (/ (/ t_0 (* (* x (* c s)) (* x c))) s))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 13000000000.0) {
		tmp = t_0 / pow((c * (x * s)), 2.0);
	} else {
		tmp = (t_0 / ((x * (c * s)) * (x * c))) / 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) :: tmp
    t_0 = cos((x + x))
    if (x <= 13000000000.0d0) then
        tmp = t_0 / ((c * (x * s)) ** 2.0d0)
    else
        tmp = (t_0 / ((x * (c * s)) * (x * c))) / s
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 13000000000.0) {
		tmp = t_0 / Math.pow((c * (x * s)), 2.0);
	} else {
		tmp = (t_0 / ((x * (c * s)) * (x * c))) / s;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 13000000000.0:
		tmp = t_0 / math.pow((c * (x * s)), 2.0)
	else:
		tmp = (t_0 / ((x * (c * s)) * (x * c))) / s
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 13000000000.0)
		tmp = Float64(t_0 / (Float64(c * Float64(x * s)) ^ 2.0));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(x * Float64(c * s)) * Float64(x * c))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 13000000000.0)
		tmp = t_0 / ((c * (x * s)) ^ 2.0);
	else
		tmp = (t_0 / ((x * (c * s)) * (x * c))) / s;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 13000000000.0], N[(t$95$0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3e10

   1. Initial program 68.9%

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

      1. associate-/r* 68.3%

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

      2. remove-double-neg 68.3%

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

      3. distribute-lft-neg-out 68.3%

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

      4. distribute-lft-neg-out 68.3%

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

      5. distribute-rgt-neg-out 68.3%

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

      6. associate-/r* 68.9%

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

      7. *-commutative 68.9%

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

      8. associate-*r* 68.4%

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

      9. associate-*r* 68.9%

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

      10. associate-*r* 63.2%

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

      11. sqr-neg 63.2%

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

      12. associate-*r* 68.9%

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

      13. *-commutative 68.9%

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

      14. unpow2 68.9%

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

      15. sqr-neg 68.9%

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

      16. unpow2 68.9%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in x around inf 63.2%

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

      1. count-2 63.2%

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

      2. *-lft-identity 63.2%

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

      3. associate-*r* 63.2%

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

      4. unpow2 63.2%

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

      5. unpow2 63.2%

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

      6. associate-*r* 70.5%

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

      7. unpow2 70.5%

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

      8. *-commutative 70.5%

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

      9. times-frac 69.5%

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

      10. associate-*r* 62.2%

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

      11. swap-sqr 79.7%

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

      12. unpow2 79.7%

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

      13. associate-/r* 79.7%

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

      14. times-frac 89.4%

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

      15. *-commutative 89.4%

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

      16. associate-/l* 89.4%

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

   6. Simplified 95.3%

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

   if 1.3e10 < x

   1. Initial program 71.5%

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

      1. associate-/r* 68.0%

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

      2. remove-double-neg 68.0%

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

      3. distribute-lft-neg-out 68.0%

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

      4. distribute-lft-neg-out 68.0%

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

      5. distribute-rgt-neg-out 68.0%

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

      6. associate-/r* 71.5%

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

      7. *-commutative 71.5%

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

      8. associate-*r* 69.8%

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

      9. associate-*r* 71.5%

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

      10. associate-*r* 66.1%

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

      11. sqr-neg 66.1%

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

      12. associate-*r* 71.5%

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

      13. *-commutative 71.5%

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

      14. unpow2 71.5%

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

      15. sqr-neg 71.5%

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

      16. unpow2 71.5%

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

   3. Simplified 69.9%

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

      1. associate-/r* 71.0%

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

      2. associate-*l* 74.8%

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

      3. associate-/r* 78.0%

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

      4. associate-*r* 86.0%

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

      5. associate-*r* 91.7%

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

      6. associate-/r* 93.2%

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

      7. div-inv 93.2%

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

      8. associate-/l/ 93.2%

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

      9. count-2 93.2%

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

      10. *-commutative 93.2%

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

   5. Applied egg-rr 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-rgt-identity 93.2%

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

      3. associate-/l/ 93.2%

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

   7. Simplified 93.2%

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

      1. add-sqr-sqrt_binary64 40.4%

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

   9. Applied rewrite-once 40.4%

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

       1. rem-square-sqrt 95.4%

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

       2. associate-*r* 96.1%

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

   11. Simplified 93.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 13000000000:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \]

Alternative 3: 87.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;s \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{elif}\;s \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (if (<= s 7.5e-14)
     (/ t_0 (* c (* x (* c (* s (* x s))))))
     (if (<= s 3.4e+201)
       (/ t_0 (* x (* c (* s (* x (* c s))))))
       (/ 1.0 (pow (* c (* x s)) 2.0))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (s <= 7.5e-14) {
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	} else if (s <= 3.4e+201) {
		tmp = t_0 / (x * (c * (s * (x * (c * 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) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if (s <= 7.5d-14) then
        tmp = t_0 / (c * (x * (c * (s * (x * s)))))
    else if (s <= 3.4d+201) then
        tmp = t_0 / (x * (c * (s * (x * (c * 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 t_0 = Math.cos((2.0 * x));
	double tmp;
	if (s <= 7.5e-14) {
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	} else if (s <= 3.4e+201) {
		tmp = t_0 / (x * (c * (s * (x * (c * s)))));
	} else {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if s <= 7.5e-14:
		tmp = t_0 / (c * (x * (c * (s * (x * s)))))
	elif s <= 3.4e+201:
		tmp = t_0 / (x * (c * (s * (x * (c * s)))))
	else:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (s <= 7.5e-14)
		tmp = Float64(t_0 / Float64(c * Float64(x * Float64(c * Float64(s * Float64(x * s))))));
	elseif (s <= 3.4e+201)
		tmp = Float64(t_0 / Float64(x * Float64(c * Float64(s * Float64(x * Float64(c * 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)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (s <= 7.5e-14)
		tmp = t_0 / (c * (x * (c * (s * (x * s)))));
	elseif (s <= 3.4e+201)
		tmp = t_0 / (x * (c * (s * (x * (c * s)))));
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 7.5e-14], N[(t$95$0 / N[(c * N[(x * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[s, 3.4e+201], N[(t$95$0 / N[(x * N[(c * N[(s * N[(x * N[(c * s), $MachinePrecision]), $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 s < 7.4999999999999996e-14

   1. Initial program 67.0%

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

      1. associate-/r* 65.9%

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

      2. remove-double-neg 65.9%

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

      3. distribute-lft-neg-out 65.9%

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

      4. distribute-lft-neg-out 65.9%

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

      5. distribute-rgt-neg-out 65.9%

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

      6. associate-/r* 67.0%

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

      7. associate-*r* 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*r* 66.1%

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

      10. unpow2 66.1%

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

      11. associate-*r* 73.1%

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

      12. *-commutative 73.1%

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

      13. associate-*l* 65.0%

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

      14. sqr-neg 65.0%

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

      15. associate-*r* 73.1%

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

      16. *-commutative 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in x around 0 67.0%

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

      1. unpow2 67.0%

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

      2. *-commutative 67.0%

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

      3. unpow2 67.0%

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

      4. associate-*l* 74.4%

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

      5. *-commutative 74.4%

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

      6. associate-*r* 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in x around 0 59.9%

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

      1. unpow2 59.9%

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

      2. associate-*r* 67.0%

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

      3. unpow2 67.0%

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

      4. associate-*r* 73.5%

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

      5. associate-*l* 72.1%

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

      6. unpow2 72.1%

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

      7. associate-*r* 79.4%

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

      8. associate-*l* 81.7%

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

   9. Simplified 81.7%

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

   if 7.4999999999999996e-14 < s < 3.4e201

   1. Initial program 80.4%

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

      1. associate-/r* 80.4%

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

      2. remove-double-neg 80.4%

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

      3. distribute-lft-neg-out 80.4%

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

      4. distribute-lft-neg-out 80.4%

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

      5. distribute-rgt-neg-out 80.4%

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

      6. associate-/r* 80.4%

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

      7. associate-*r* 80.4%

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

      8. *-commutative 80.4%

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

      9. associate-*r* 80.6%

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

      10. unpow2 80.6%

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

      11. associate-*r* 88.7%

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

      12. *-commutative 88.7%

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

      13. associate-*l* 82.8%

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

      14. sqr-neg 82.8%

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

      15. associate-*r* 88.7%

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

      16. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. unpow2 80.4%

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

      2. *-commutative 80.4%

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

      3. unpow2 80.4%

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

      4. associate-*l* 84.9%

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

      5. *-commutative 84.9%

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

      6. associate-*r* 88.9%

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

   6. Simplified 88.9%

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

   7. Taylor expanded in c around 0 84.9%

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

      1. *-commutative 84.9%

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

      2. unpow2 84.9%

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

      3. associate-*r* 88.9%

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

      4. associate-*l* 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

   9. Simplified 98.4%

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

   if 3.4e201 < s

   1. Initial program 65.8%

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

      1. associate-/r* 60.8%

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

      2. remove-double-neg 60.8%

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

      3. distribute-lft-neg-out 60.8%

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

      4. distribute-lft-neg-out 60.8%

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

      5. distribute-rgt-neg-out 60.8%

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

      6. associate-/r* 65.8%

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

      7. *-commutative 65.8%

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

      8. associate-*r* 65.8%

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

      9. associate-*r* 65.8%

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

      10. associate-*r* 65.3%

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

      11. sqr-neg 65.3%

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

      12. associate-*r* 65.8%

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

      13. *-commutative 65.8%

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

      14. unpow2 65.8%

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

      15. sqr-neg 65.8%

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

      16. unpow2 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in x around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. *-commutative 65.3%

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

      3. unpow2 65.3%

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

      4. unpow2 65.3%

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

      5. associate-*r* 65.8%

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

      6. unpow2 65.8%

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

      7. *-commutative 65.8%

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

      8. associate-*r* 65.3%

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

      9. *-commutative 65.3%

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

      10. swap-sqr 70.6%

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

      11. swap-sqr 99.7%

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

      12. unpow2 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{elif}\;s \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 4: 87.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.1e+145)
   (/ (cos (* 2.0 x)) (* c (* x (* c (* s (* x s))))))
   (/ 1.0 (pow (* c (* x s)) 2.0))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.1e+145) {
		tmp = cos((2.0 * x)) / (c * (x * (c * (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 (s <= 1.1d+145) then
        tmp = cos((2.0d0 * x)) / (c * (x * (c * (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 (s <= 1.1e+145) {
		tmp = Math.cos((2.0 * x)) / (c * (x * (c * (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 s <= 1.1e+145:
		tmp = math.cos((2.0 * x)) / (c * (x * (c * (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 (s <= 1.1e+145)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(c * Float64(x * Float64(c * 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 (s <= 1.1e+145)
		tmp = cos((2.0 * x)) / (c * (x * (c * (s * (x * s)))));
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.1e+145], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(x * N[(c * N[(s * N[(x * s), $MachinePrecision]), $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 2 regimes

2. if s < 1.10000000000000004e145

   1. Initial program 70.9%

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

      1. associate-/r* 70.0%

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

      2. remove-double-neg 70.0%

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

      3. distribute-lft-neg-out 70.0%

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

      4. distribute-lft-neg-out 70.0%

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

      5. distribute-rgt-neg-out 70.0%

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

      6. associate-/r* 70.9%

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

      7. associate-*r* 70.9%

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

      8. *-commutative 70.9%

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

      9. associate-*r* 70.2%

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

      10. unpow2 70.2%

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

      11. associate-*r* 77.2%

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

      12. *-commutative 77.2%

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

      13. associate-*l* 69.3%

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

      14. sqr-neg 69.3%

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

      15. associate-*r* 77.2%

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

      16. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in x around 0 70.9%

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

      1. unpow2 70.9%

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

      2. *-commutative 70.9%

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

      3. unpow2 70.9%

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

      4. associate-*l* 77.8%

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

      5. *-commutative 77.8%

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

      6. associate-*r* 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in x around 0 64.7%

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

      1. unpow2 64.7%

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

      2. associate-*r* 70.9%

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

      3. unpow2 70.9%

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

      4. associate-*r* 76.2%

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

      5. associate-*l* 75.1%

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

      6. unpow2 75.1%

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

      7. associate-*r* 82.0%

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

      8. associate-*l* 83.9%

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

   9. Simplified 83.9%

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

   if 1.10000000000000004e145 < s

   1. Initial program 57.9%

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

      1. associate-/r* 54.3%

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

      2. remove-double-neg 54.3%

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

      3. distribute-lft-neg-out 54.3%

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

      4. distribute-lft-neg-out 54.3%

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

      5. distribute-rgt-neg-out 54.3%

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

      6. associate-/r* 57.9%

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

      7. *-commutative 57.9%

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

      8. associate-*r* 57.8%

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

      9. associate-*r* 57.9%

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

      10. associate-*r* 57.4%

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

      11. sqr-neg 57.4%

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

      12. associate-*r* 57.9%

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

      13. *-commutative 57.9%

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

      14. unpow2 57.9%

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

      15. sqr-neg 57.9%

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

      16. unpow2 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in x around 0 57.4%

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

      1. *-commutative 57.4%

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

      2. *-commutative 57.4%

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

      3. unpow2 57.4%

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

      4. unpow2 57.4%

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

      5. associate-*r* 57.9%

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

      6. unpow2 57.9%

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

      7. *-commutative 57.9%

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

      8. associate-*r* 57.4%

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

      9. *-commutative 57.4%

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

      10. swap-sqr 68.3%

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

      11. swap-sqr 89.8%

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

      12. unpow2 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 5: 86.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 3 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{{t_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_0 \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 3e-37)
     (/ 1.0 (pow t_0 2.0))
     (/ (/ (cos (+ x x)) (* t_0 (* x c))) s))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 3e-37) {
		tmp = 1.0 / pow(t_0, 2.0);
	} else {
		tmp = (cos((x + x)) / (t_0 * (x * c))) / 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) :: tmp
    t_0 = c * (x * s)
    if (x <= 3d-37) then
        tmp = 1.0d0 / (t_0 ** 2.0d0)
    else
        tmp = (cos((x + x)) / (t_0 * (x * c))) / s
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 3e-37) {
		tmp = 1.0 / Math.pow(t_0, 2.0);
	} else {
		tmp = (Math.cos((x + x)) / (t_0 * (x * c))) / s;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 3e-37:
		tmp = 1.0 / math.pow(t_0, 2.0)
	else:
		tmp = (math.cos((x + x)) / (t_0 * (x * c))) / s
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 3e-37)
		tmp = Float64(1.0 / (t_0 ^ 2.0));
	else
		tmp = Float64(Float64(cos(Float64(x + x)) / Float64(t_0 * Float64(x * c))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 3e-37)
		tmp = 1.0 / (t_0 ^ 2.0);
	else
		tmp = (cos((x + x)) / (t_0 * (x * c))) / s;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3e-37], N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e-37

   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. associate-/r* 67.0%

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

      2. remove-double-neg 67.0%

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

      3. distribute-lft-neg-out 67.0%

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

      4. distribute-lft-neg-out 67.0%

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

      5. distribute-rgt-neg-out 67.0%

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

      6. associate-/r* 67.6%

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

      7. *-commutative 67.6%

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

      8. associate-*r* 67.1%

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

      9. associate-*r* 67.6%

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

      10. associate-*r* 61.6%

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

      11. sqr-neg 61.6%

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

      12. associate-*r* 67.6%

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

      13. *-commutative 67.6%

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

      14. unpow2 67.6%

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

      15. sqr-neg 67.6%

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

      16. unpow2 67.6%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. *-commutative 57.7%

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

      3. unpow2 57.7%

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

      4. unpow2 57.7%

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

      5. associate-*r* 60.7%

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

      6. unpow2 60.7%

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

      7. *-commutative 60.7%

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

      8. associate-*r* 57.7%

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

      9. *-commutative 57.7%

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

      10. swap-sqr 66.7%

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

      11. swap-sqr 81.1%

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

      12. unpow2 81.1%

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

   6. Simplified 81.1%

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

   if 3e-37 < x

   1. Initial program 74.6%

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

      1. associate-/r* 71.6%

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

      2. remove-double-neg 71.6%

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

      3. distribute-lft-neg-out 71.6%

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

      4. distribute-lft-neg-out 71.6%

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

      5. distribute-rgt-neg-out 71.6%

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

      6. associate-/r* 74.6%

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

      7. *-commutative 74.6%

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

      8. associate-*r* 73.2%

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

      9. associate-*r* 74.6%

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

      10. associate-*r* 70.0%

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

      11. sqr-neg 70.0%

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

      12. associate-*r* 74.6%

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

      13. *-commutative 74.6%

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

      14. unpow2 74.6%

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

      15. sqr-neg 74.6%

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

      16. unpow2 74.6%

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

   3. Simplified 73.3%

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

      1. associate-/r* 74.1%

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

      2. associate-*l* 77.4%

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

      3. associate-/r* 80.1%

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

      4. associate-*r* 86.8%

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

      5. associate-*r* 93.0%

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

      6. associate-/r* 94.2%

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

      7. div-inv 94.2%

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

      8. associate-/l/ 94.2%

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

      9. count-2 94.2%

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

      10. *-commutative 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. associate-*r/ 94.2%

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

      2. *-rgt-identity 94.2%

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

      3. associate-/l/ 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \]

Alternative 6: 86.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 5e-23)
   (/ 1.0 (pow (* c (* x s)) 2.0))
   (/ (/ (cos (+ x x)) (* (* x (* c s)) (* x c))) s)))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-23) {
		tmp = 1.0 / pow((c * (x * s)), 2.0);
	} else {
		tmp = (cos((x + x)) / ((x * (c * s)) * (x * c))) / s;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 5d-23) then
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    else
        tmp = (cos((x + x)) / ((x * (c * s)) * (x * c))) / s
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-23) {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	} else {
		tmp = (Math.cos((x + x)) / ((x * (c * s)) * (x * c))) / s;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 5e-23:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	else:
		tmp = (math.cos((x + x)) / ((x * (c * s)) * (x * c))) / s
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 5e-23)
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	else
		tmp = Float64(Float64(cos(Float64(x + x)) / Float64(Float64(x * Float64(c * s)) * Float64(x * c))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5e-23)
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	else
		tmp = (cos((x + x)) / ((x * (c * s)) * (x * c))) / s;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 5e-23], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-23

   1. Initial program 67.7%

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

      1. associate-/r* 67.2%

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

      2. remove-double-neg 67.2%

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

      3. distribute-lft-neg-out 67.2%

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

      4. distribute-lft-neg-out 67.2%

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

      5. distribute-rgt-neg-out 67.2%

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

      6. associate-/r* 67.7%

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

      7. *-commutative 67.7%

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

      8. associate-*r* 67.2%

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

      9. associate-*r* 67.7%

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

      10. associate-*r* 61.9%

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

      11. sqr-neg 61.9%

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

      12. associate-*r* 67.7%

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

      13. *-commutative 67.7%

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

      14. unpow2 67.7%

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

      15. sqr-neg 67.7%

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

      16. unpow2 67.7%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in x around 0 58.0%

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

      1. *-commutative 58.0%

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

      2. *-commutative 58.0%

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

      3. unpow2 58.0%

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

      4. unpow2 58.0%

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

      5. associate-*r* 61.0%

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

      6. unpow2 61.0%

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

      7. *-commutative 61.0%

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

      8. associate-*r* 58.0%

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

      9. *-commutative 58.0%

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

      10. swap-sqr 66.8%

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

      11. swap-sqr 81.5%

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

      12. unpow2 81.5%

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

   6. Simplified 81.5%

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

   if 5.0000000000000002e-23 < x

   1. Initial program 74.6%

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

      1. associate-/r* 71.5%

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

      2. remove-double-neg 71.5%

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

      3. distribute-lft-neg-out 71.5%

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

      4. distribute-lft-neg-out 71.5%

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

      5. distribute-rgt-neg-out 71.5%

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

      6. associate-/r* 74.6%

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

      7. *-commutative 74.6%

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

      8. associate-*r* 73.1%

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

      9. associate-*r* 74.6%

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

      10. associate-*r* 69.8%

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

      11. sqr-neg 69.8%

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

      12. associate-*r* 74.6%

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

      13. *-commutative 74.6%

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

      14. unpow2 74.6%

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

      15. sqr-neg 74.6%

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

      16. unpow2 74.6%

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

   3. Simplified 73.2%

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

      1. associate-/r* 74.1%

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

      2. associate-*l* 77.5%

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

      3. associate-/r* 80.4%

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

      4. associate-*r* 87.5%

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

      5. associate-*r* 92.6%

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

      6. associate-/r* 93.9%

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

      7. div-inv 93.9%

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

      8. associate-/l/ 93.9%

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

      9. count-2 93.9%

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

      10. *-commutative 93.9%

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

   5. Applied egg-rr 93.9%

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

      1. associate-*r/ 93.9%

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

      2. *-rgt-identity 93.9%

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

      3. associate-/l/ 93.9%

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

   7. Simplified 93.9%

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

      1. add-sqr-sqrt_binary64 42.2%

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

   9. Applied rewrite-once 42.2%

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

       1. rem-square-sqrt 95.8%

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

       2. associate-*r* 96.4%

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

   11. Simplified 94.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)}}{s}\\ \end{array} \]

Alternative 7: 54.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.75 \cdot 10^{-120}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.75e-120)
   (/ (cos (+ x x)) (* (* x (* c c)) (* s s)))
   (/ 1.0 (pow (* c (* x s)) 2.0))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.75e-120) {
		tmp = cos((x + x)) / ((x * (c * c)) * (s * 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 (s <= 1.75d-120) then
        tmp = cos((x + x)) / ((x * (c * c)) * (s * s))
    else
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.75e-120) {
		tmp = Math.cos((x + x)) / ((x * (c * c)) * (s * s));
	} else {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.75e-120:
		tmp = math.cos((x + x)) / ((x * (c * c)) * (s * s))
	else:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.75e-120)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(x * Float64(c * c)) * Float64(s * s)));
	else
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 1.75e-120)
		tmp = cos((x + x)) / ((x * (c * c)) * (s * s));
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.75e-120], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.75e-120

   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. 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. remove-double-neg 64.3%

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

      3. distribute-lft-neg-out 64.3%

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

      4. distribute-lft-neg-out 64.3%

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

      5. distribute-rgt-neg-out 64.3%

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

      6. associate-/r* 65.5%

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

      7. *-commutative 65.5%

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

      8. associate-*r* 64.9%

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

      9. associate-*r* 65.5%

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

      10. associate-*r* 58.2%

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

      11. sqr-neg 58.2%

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

      12. associate-*r* 65.5%

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

      13. *-commutative 65.5%

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

      14. unpow2 65.5%

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

      15. sqr-neg 65.5%

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

      16. unpow2 65.5%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in x around inf 58.2%

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

      1. associate-*r* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. associate-*r* 63.4%

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

      5. unpow2 63.4%

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

      6. *-commutative 63.4%

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

      7. associate-*r* 72.4%

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

      8. associate-/r* 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-*r* 65.0%

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

      11. unpow2 65.0%

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

      12. unpow2 65.0%

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

      13. associate-*r* 65.5%

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

      14. unpow2 65.5%

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

      15. *-commutative 65.5%

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

      16. unpow2 65.5%

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

      17. associate-/r* 65.6%

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

      18. count-2 65.6%

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

   6. Simplified 70.3%

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

   8. Applied egg-rr 8.7%

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

      1. distribute-lft-neg-in 8.7%

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

      2. metadata-eval 8.7%

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

      3. log-pow 38.1%

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

      4. unpow2 38.1%

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

      5. log-prod 8.7%

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

      6. exp-sum 8.7%

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

      7. rem-exp-log 8.7%

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

      8. rem-exp-log 38.1%

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

   10. Simplified 38.1%

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

   if 1.75e-120 < s

   1. Initial program 76.5%

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

      1. associate-/r* 75.5%

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

      2. remove-double-neg 75.5%

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

      3. distribute-lft-neg-out 75.5%

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

      4. distribute-lft-neg-out 75.5%

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

      5. distribute-rgt-neg-out 75.5%

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

      6. associate-/r* 76.5%

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

      7. *-commutative 76.5%

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

      8. associate-*r* 75.6%

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

      9. associate-*r* 76.5%

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

      10. associate-*r* 74.2%

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

      11. sqr-neg 74.2%

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

      12. associate-*r* 76.5%

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

      13. *-commutative 76.5%

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

      14. unpow2 76.5%

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

      15. sqr-neg 76.5%

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

      16. unpow2 76.5%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in x around 0 70.1%

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

      1. *-commutative 70.1%

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

      2. *-commutative 70.1%

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

      3. unpow2 70.1%

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

      4. unpow2 70.1%

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

      5. associate-*r* 71.6%

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

      6. unpow2 71.6%

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

      7. *-commutative 71.6%

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

      8. associate-*r* 70.1%

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

      9. *-commutative 70.1%

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

      10. swap-sqr 74.9%

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

      11. swap-sqr 85.5%

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

      12. unpow2 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.75 \cdot 10^{-120}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 8: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.65 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 4.65e-122)
   (/ (/ (cos (+ x x)) (* s (* x c))) s)
   (/ 1.0 (pow (* c (* x s)) 2.0))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.65e-122) {
		tmp = (cos((x + x)) / (s * (x * c))) / 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 (s <= 4.65d-122) then
        tmp = (cos((x + x)) / (s * (x * c))) / s
    else
        tmp = 1.0d0 / ((c * (x * s)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.65e-122) {
		tmp = (Math.cos((x + x)) / (s * (x * c))) / s;
	} else {
		tmp = 1.0 / Math.pow((c * (x * s)), 2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 4.65e-122:
		tmp = (math.cos((x + x)) / (s * (x * c))) / s
	else:
		tmp = 1.0 / math.pow((c * (x * s)), 2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 4.65e-122)
		tmp = Float64(Float64(cos(Float64(x + x)) / Float64(s * Float64(x * c))) / s);
	else
		tmp = Float64(1.0 / (Float64(c * Float64(x * s)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 4.65e-122)
		tmp = (cos((x + x)) / (s * (x * c))) / s;
	else
		tmp = 1.0 / ((c * (x * s)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 4.65e-122], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 4.6499999999999998e-122

   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. 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. remove-double-neg 64.3%

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

      3. distribute-lft-neg-out 64.3%

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

      4. distribute-lft-neg-out 64.3%

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

      5. distribute-rgt-neg-out 64.3%

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

      6. associate-/r* 65.5%

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

      7. *-commutative 65.5%

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

      8. associate-*r* 64.9%

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

      9. associate-*r* 65.5%

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

      10. associate-*r* 58.2%

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

      11. sqr-neg 58.2%

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

      12. associate-*r* 65.5%

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

      13. *-commutative 65.5%

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

      14. unpow2 65.5%

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

      15. sqr-neg 65.5%

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

      16. unpow2 65.5%

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

   3. Simplified 65.6%

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

      1. associate-/r* 65.5%

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

      2. associate-*l* 72.6%

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

      3. associate-/r* 74.8%

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

      4. associate-*r* 79.9%

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

      5. associate-*r* 88.7%

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

      6. associate-/r* 88.0%

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

      7. div-inv 88.1%

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

      8. associate-/l/ 88.1%

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

      9. count-2 88.1%

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

      10. *-commutative 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. associate-/l/ 87.7%

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

   7. Simplified 87.7%

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

   9. Applied egg-rr 43.0%

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

       1. associate-/l* 39.1%

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

       2. associate-/r/ 43.0%

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

       3. *-inverses 43.0%

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

       4. *-lft-identity 43.0%

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

   11. Simplified 43.0%

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

   if 4.6499999999999998e-122 < s

   1. Initial program 76.5%

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

      1. associate-/r* 75.5%

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

      2. remove-double-neg 75.5%

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

      3. distribute-lft-neg-out 75.5%

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

      4. distribute-lft-neg-out 75.5%

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

      5. distribute-rgt-neg-out 75.5%

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

      6. associate-/r* 76.5%

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

      7. *-commutative 76.5%

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

      8. associate-*r* 75.6%

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

      9. associate-*r* 76.5%

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

      10. associate-*r* 74.2%

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

      11. sqr-neg 74.2%

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

      12. associate-*r* 76.5%

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

      13. *-commutative 76.5%

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

      14. unpow2 76.5%

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

      15. sqr-neg 76.5%

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

      16. unpow2 76.5%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in x around 0 70.1%

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

      1. *-commutative 70.1%

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

      2. *-commutative 70.1%

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

      3. unpow2 70.1%

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

      4. unpow2 70.1%

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

      5. associate-*r* 71.6%

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

      6. unpow2 71.6%

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

      7. *-commutative 71.6%

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

      8. associate-*r* 70.1%

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

      9. *-commutative 70.1%

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

      10. swap-sqr 74.9%

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

      11. swap-sqr 85.5%

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

      12. unpow2 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.65 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 9: 78.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. associate-/r* 68.3%

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

   2. remove-double-neg 68.3%

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

   3. distribute-lft-neg-out 68.3%

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

   4. distribute-lft-neg-out 68.3%

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

   5. distribute-rgt-neg-out 68.3%

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

   6. associate-/r* 69.5%

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

   7. *-commutative 69.5%

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

   8. associate-*r* 68.7%

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

   9. associate-*r* 69.5%

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

   10. associate-*r* 63.9%

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

   11. sqr-neg 63.9%

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

   12. associate-*r* 69.5%

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

   13. *-commutative 69.5%

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

   14. unpow2 69.5%

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

   15. sqr-neg 69.5%

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

   16. unpow2 69.5%

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

3. Simplified 69.5%

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

4. Taylor expanded in x around 0 58.1%

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

   1. *-commutative 58.1%

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

   2. *-commutative 58.1%

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

   3. unpow2 58.1%

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

   4. unpow2 58.1%

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

   5. associate-*r* 60.9%

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

   6. unpow2 60.9%

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

   7. *-commutative 60.9%

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

   8. associate-*r* 58.1%

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

   9. *-commutative 58.1%

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

   10. swap-sqr 66.3%

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

   11. swap-sqr 77.9%

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

   12. unpow2 77.9%

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

6. Simplified 77.9%

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

7. Final simplification 77.9%

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

Alternative 10: 63.6% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}\right) \cdot \frac{1}{c \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 8.5e-122)
   (* (- (/ 1.0 (* c (* x x))) (/ 2.0 c)) (/ 1.0 (* c (* s s))))
   (/ 1.0 (* c (* x (* s (* x (* c s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 8.5e-122) {
		tmp = ((1.0 / (c * (x * x))) - (2.0 / c)) * (1.0 / (c * (s * s)));
	} else {
		tmp = 1.0 / (c * (x * (s * (x * (c * s)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 8.5d-122) then
        tmp = ((1.0d0 / (c * (x * x))) - (2.0d0 / c)) * (1.0d0 / (c * (s * s)))
    else
        tmp = 1.0d0 / (c * (x * (s * (x * (c * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 8.5e-122) {
		tmp = ((1.0 / (c * (x * x))) - (2.0 / c)) * (1.0 / (c * (s * s)));
	} else {
		tmp = 1.0 / (c * (x * (s * (x * (c * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 8.5e-122:
		tmp = ((1.0 / (c * (x * x))) - (2.0 / c)) * (1.0 / (c * (s * s)))
	else:
		tmp = 1.0 / (c * (x * (s * (x * (c * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 8.5e-122)
		tmp = Float64(Float64(Float64(1.0 / Float64(c * Float64(x * x))) - Float64(2.0 / c)) * Float64(1.0 / Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(x * Float64(s * Float64(x * Float64(c * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 8.5e-122)
		tmp = ((1.0 / (c * (x * x))) - (2.0 / c)) * (1.0 / (c * (s * s)));
	else
		tmp = 1.0 / (c * (x * (s * (x * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 8.5e-122], N[(N[(N[(1.0 / N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(x * N[(s * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 8.50000000000000003e-122

   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. 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. remove-double-neg 64.3%

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

      3. distribute-lft-neg-out 64.3%

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

      4. distribute-lft-neg-out 64.3%

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

      5. distribute-rgt-neg-out 64.3%

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

      6. associate-/r* 65.5%

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

      7. *-commutative 65.5%

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

      8. associate-*r* 64.9%

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

      9. associate-*r* 65.5%

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

      10. associate-*r* 58.2%

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

      11. sqr-neg 58.2%

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

      12. associate-*r* 65.5%

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

      13. *-commutative 65.5%

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

      14. unpow2 65.5%

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

      15. sqr-neg 65.5%

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

      16. unpow2 65.5%

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

   3. Simplified 65.6%

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

      1. associate-/r* 65.5%

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

      2. associate-*r* 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*r* 71.4%

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

      5. associate-*r* 75.8%

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

      6. associate-/r* 72.5%

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

      7. div-inv 72.1%

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

      8. cos-2 72.1%

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

      9. cos-sum 72.1%

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

      10. *-commutative 72.1%

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

   5. Applied egg-rr 72.1%

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

   6. Taylor expanded in x around 0 54.3%

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

      1. unpow2 54.3%

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

      2. associate-*r/ 54.3%

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

      3. metadata-eval 54.3%

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

   8. Simplified 54.3%

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

   if 8.50000000000000003e-122 < s

   1. Initial program 76.5%

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

      1. associate-/r* 75.5%

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

      2. remove-double-neg 75.5%

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

      3. distribute-lft-neg-out 75.5%

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

      4. distribute-lft-neg-out 75.5%

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

      5. distribute-rgt-neg-out 75.5%

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

      6. associate-/r* 76.5%

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

      7. *-commutative 76.5%

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

      8. associate-*r* 75.6%

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

      9. associate-*r* 76.5%

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

      10. associate-*r* 74.2%

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

      11. sqr-neg 74.2%

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

      12. associate-*r* 76.5%

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

      13. *-commutative 76.5%

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

      14. unpow2 76.5%

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

      15. sqr-neg 76.5%

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

      16. unpow2 76.5%

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

   3. Simplified 76.6%

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

      1. associate-/r* 77.3%

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

      2. associate-*l* 86.5%

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

      3. associate-/r* 87.5%

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

      4. associate-*r* 89.6%

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

      5. associate-*r* 93.5%

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

      6. associate-/r* 92.5%

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

      7. div-inv 92.5%

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

      8. associate-/l/ 92.5%

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

      9. count-2 92.5%

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

      10. *-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. associate-*r/ 92.6%

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

      2. *-rgt-identity 92.6%

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

      3. associate-/l/ 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in x around 0 70.1%

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

      1. unpow2 70.1%

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

      2. associate-*r* 71.6%

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

      3. unpow2 71.6%

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

      4. associate-*r* 73.8%

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

      5. associate-*l* 73.9%

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

      6. unpow2 73.9%

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

      7. associate-*r* 81.0%

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

      8. associate-*l* 81.0%

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

      9. *-commutative 81.0%

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

      10. associate-*l* 82.2%

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

      11. *-commutative 82.2%

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

      12. associate-*r* 82.2%

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

      13. *-commutative 82.2%

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

   10. Simplified 82.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}\right) \cdot \frac{1}{c \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 11: 42.5% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 3.3e-177)
   (/ 1.0 (* (- c c) (/ s s)))
   (if (<= x 1.2e+133)
     (/ 1.0 (* (* s (* x s)) (* c c)))
     (/ 1.0 (* c (* s (* x (- s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else if (x <= 1.2e+133) {
		tmp = 1.0 / ((s * (x * s)) * (c * c));
	} else {
		tmp = 1.0 / (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) :: tmp
    if (x <= 3.3d-177) then
        tmp = 1.0d0 / ((c - c) * (s / s))
    else if (x <= 1.2d+133) then
        tmp = 1.0d0 / ((s * (x * s)) * (c * c))
    else
        tmp = 1.0d0 / (c * (s * (x * -s)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else if (x <= 1.2e+133) {
		tmp = 1.0 / ((s * (x * s)) * (c * c));
	} else {
		tmp = 1.0 / (c * (s * (x * -s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 3.3e-177:
		tmp = 1.0 / ((c - c) * (s / s))
	elif x <= 1.2e+133:
		tmp = 1.0 / ((s * (x * s)) * (c * c))
	else:
		tmp = 1.0 / (c * (s * (x * -s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 3.3e-177)
		tmp = Float64(1.0 / Float64(Float64(c - c) * Float64(s / s)));
	elseif (x <= 1.2e+133)
		tmp = Float64(1.0 / Float64(Float64(s * Float64(x * s)) * Float64(c * c)));
	else
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(x * Float64(-s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 3.3e-177)
		tmp = 1.0 / ((c - c) * (s / s));
	elseif (x <= 1.2e+133)
		tmp = 1.0 / ((s * (x * s)) * (c * c));
	else
		tmp = 1.0 / (c * (s * (x * -s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 3.3e-177], N[(1.0 / N[(N[(c - c), $MachinePrecision] * N[(s / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+133], N[(1.0 / N[(N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(s * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.3e-177

   1. Initial program 67.2%

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

      1. associate-/r* 66.6%

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

      2. remove-double-neg 66.6%

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

      3. distribute-lft-neg-out 66.6%

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

      4. distribute-lft-neg-out 66.6%

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

      5. distribute-rgt-neg-out 66.6%

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

      6. associate-/r* 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 66.7%

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

      9. associate-*r* 67.2%

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

      10. associate-*r* 60.5%

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

      11. sqr-neg 60.5%

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

      12. associate-*r* 67.2%

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

      13. *-commutative 67.2%

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

      14. unpow2 67.2%

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

      15. sqr-neg 67.2%

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

      16. unpow2 67.2%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in x around 0 56.0%

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

      1. unpow2 56.0%

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

      2. *-commutative 56.0%

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

      3. unpow2 56.0%

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

      4. unpow2 56.0%

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

      5. associate-*r* 59.4%

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

      6. associate-*r* 58.8%

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

      7. *-commutative 58.8%

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

      8. associate-*r* 59.5%

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

      9. associate-*r* 58.8%

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

      10. *-commutative 58.8%

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

      11. *-commutative 58.8%

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

      12. associate-*r* 61.2%

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

   6. Simplified 61.2%

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

   8. Applied egg-rr 28.6%

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

   10. Applied egg-rr 35.0%

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

   if 3.3e-177 < x < 1.1999999999999999e133

   1. Initial program 75.2%

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

      1. associate-/r* 75.1%

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

      2. remove-double-neg 75.1%

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

      3. distribute-lft-neg-out 75.1%

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

      4. distribute-lft-neg-out 75.1%

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

      5. distribute-rgt-neg-out 75.1%

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

      6. associate-/r* 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-*r* 73.4%

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

      9. associate-*r* 75.2%

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

      10. associate-*r* 75.2%

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

      11. sqr-neg 75.2%

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

      12. associate-*r* 75.2%

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

      13. *-commutative 75.2%

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

      14. unpow2 75.2%

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

      15. sqr-neg 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 64.4%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 64.4%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 64.3%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 64.3%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 66.1%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 26.6%

      \[\leadsto \frac{1}{\color{blue}{e^{--2 \cdot \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-neg-in 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(--2\right) \cdot \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      2. metadata-eval 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{2} \cdot \log c} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      3. count-2 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\log c + \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      4. prod-exp 26.6%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\log c} \cdot e^{\log c}\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      5. rem-exp-log 26.6%

         \[\leadsto \frac{1}{\left(\color{blue}{c} \cdot e^{\log c}\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      6. rem-exp-log 55.0%

         \[\leadsto \frac{1}{\left(c \cdot \color{blue}{c}\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

   10. Simplified 55.0%

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

   if 1.1999999999999999e133 < x

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 64.4%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 61.0%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 61.0%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 58.0%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 62.8%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 62.4%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 62.8%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 62.8%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 69.4%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 59.7%

      \[\leadsto \frac{1}{\color{blue}{\left(-c\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \end{array} \]

Alternative 12: 42.5% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{t_0 \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot \frac{c}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x s))))
   (if (<= x 3.2e-177)
     (/ 1.0 (* (- c c) (/ s s)))
     (if (<= x 1.2e+133) (/ 1.0 (* t_0 (* c c))) (/ 1.0 (* t_0 (/ c -2.0)))))))

C

double code(double x, double c, double s) {
	double t_0 = s * (x * s);
	double tmp;
	if (x <= 3.2e-177) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else if (x <= 1.2e+133) {
		tmp = 1.0 / (t_0 * (c * c));
	} else {
		tmp = 1.0 / (t_0 * (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 = s * (x * s)
    if (x <= 3.2d-177) then
        tmp = 1.0d0 / ((c - c) * (s / s))
    else if (x <= 1.2d+133) then
        tmp = 1.0d0 / (t_0 * (c * c))
    else
        tmp = 1.0d0 / (t_0 * (c / (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = s * (x * s);
	double tmp;
	if (x <= 3.2e-177) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else if (x <= 1.2e+133) {
		tmp = 1.0 / (t_0 * (c * c));
	} else {
		tmp = 1.0 / (t_0 * (c / -2.0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = s * (x * s)
	tmp = 0
	if x <= 3.2e-177:
		tmp = 1.0 / ((c - c) * (s / s))
	elif x <= 1.2e+133:
		tmp = 1.0 / (t_0 * (c * c))
	else:
		tmp = 1.0 / (t_0 * (c / -2.0))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(s * Float64(x * s))
	tmp = 0.0
	if (x <= 3.2e-177)
		tmp = Float64(1.0 / Float64(Float64(c - c) * Float64(s / s)));
	elseif (x <= 1.2e+133)
		tmp = Float64(1.0 / Float64(t_0 * Float64(c * c)));
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(c / -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = s * (x * s);
	tmp = 0.0;
	if (x <= 3.2e-177)
		tmp = 1.0 / ((c - c) * (s / s));
	elseif (x <= 1.2e+133)
		tmp = 1.0 / (t_0 * (c * c));
	else
		tmp = 1.0 / (t_0 * (c / -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.2e-177], N[(1.0 / N[(N[(c - c), $MachinePrecision] * N[(s / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+133], N[(1.0 / N[(t$95$0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * N[(c / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1999999999999998e-177

   1. Initial program 67.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 66.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 66.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 66.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 66.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 67.2%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 67.2%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 66.7%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 67.2%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 60.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 60.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 67.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 67.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 67.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 67.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 67.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 56.0%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 56.0%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 56.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 56.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 56.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 59.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 58.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 58.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 59.5%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 58.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 58.8%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 58.8%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 61.2%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 28.6%

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\frac{s}{s}}} \]

   10. Applied egg-rr 35.0%

       \[\leadsto \frac{1}{\color{blue}{\left(c - c\right)} \cdot \frac{s}{s}} \]

   if 3.1999999999999998e-177 < x < 1.1999999999999999e133

   1. Initial program 75.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.1%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 75.1%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 75.1%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 75.1%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 75.1%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 75.2%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 75.2%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 73.4%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 75.2%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 75.2%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 64.4%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 64.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 64.4%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 64.3%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 64.3%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 64.3%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 66.1%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 26.6%

      \[\leadsto \frac{1}{\color{blue}{e^{--2 \cdot \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-neg-in 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\left(--2\right) \cdot \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      2. metadata-eval 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{2} \cdot \log c} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      3. count-2 26.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\log c + \log c}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      4. prod-exp 26.6%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\log c} \cdot e^{\log c}\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      5. rem-exp-log 26.6%

         \[\leadsto \frac{1}{\left(\color{blue}{c} \cdot e^{\log c}\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

      6. rem-exp-log 55.0%

         \[\leadsto \frac{1}{\left(c \cdot \color{blue}{c}\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

   10. Simplified 55.0%

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]

   if 1.1999999999999999e133 < x

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 64.4%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 64.4%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 61.0%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 61.0%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 58.0%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 58.0%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 62.8%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 62.4%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 62.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 62.8%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 62.8%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 69.4%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 59.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{-2}} \cdot \left(s \cdot \left(s \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot s\right)\right) \cdot \frac{c}{-2}}\\ \end{array} \]

Alternative 13: 72.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* c (* x (* c (* s (* x s)))))))

C

double code(double x, double c, double s) {
	return 1.0 / (c * (x * (c * (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 * (x * (c * (s * (x * s)))))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (c * (x * (c * (s * (x * s)))));
}

Python

def code(x, c, s):
	return 1.0 / (c * (x * (c * (s * (x * s)))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(c * Float64(x * Float64(c * Float64(s * Float64(x * s))))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (c * (x * (c * (s * (x * s)))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(c * N[(x * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 68.3%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/r* 69.5%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

   7. *-commutative 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   8. associate-*r* 68.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

   9. associate-*r* 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   10. associate-*r* 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

   11. sqr-neg 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

   12. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   13. *-commutative 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   14. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   15. sqr-neg 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   16. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

3. Simplified 69.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   2. associate-*l* 77.5%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   3. associate-/r* 79.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   4. associate-*r* 83.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

   5. associate-*r* 90.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

   6. associate-/r* 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

   7. div-inv 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

   8. associate-/l/ 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   9. count-2 89.7%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   10. *-commutative 89.7%

       \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

5. Applied egg-rr 89.7%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
6. Step-by-step derivation

   1. associate-*r/ 89.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

   2. *-rgt-identity 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

   3. associate-/l/ 89.4%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

7. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

8. Taylor expanded in x around 0 58.1%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
9. Step-by-step derivation

   1. unpow2 58.1%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   2. associate-*r* 60.9%

      \[\leadsto \frac{1}{{c}^{2} \cdot \color{blue}{\left(\left({s}^{2} \cdot x\right) \cdot x\right)}} \]

   3. unpow2 60.9%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot x\right)} \]

   4. associate-*r* 64.5%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot x\right)} \]

   5. associate-*l* 64.1%

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x}} \]

   6. unpow2 64.1%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x} \]

   7. associate-*r* 70.2%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)\right)} \cdot x} \]

   8. associate-*l* 70.7%

      \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x\right)}} \]

10. Simplified 70.7%

    \[\leadsto \color{blue}{\frac{1}{c \cdot \left(\left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x\right)}} \]

11. Final simplification 70.7%

    \[\leadsto \frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)} \]

Alternative 14: 75.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* c (* x (* s (* x (* c s)))))))

C

double code(double x, double c, double s) {
	return 1.0 / (c * (x * (s * (x * (c * s)))));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / (c * (x * (s * (x * (c * s)))))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (c * (x * (s * (x * (c * s)))));
}

Python

def code(x, c, s):
	return 1.0 / (c * (x * (s * (x * (c * s)))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(c * Float64(x * Float64(s * Float64(x * Float64(c * s))))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (c * (x * (s * (x * (c * s)))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(c * N[(x * N[(s * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 68.3%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/r* 69.5%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

   7. *-commutative 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   8. associate-*r* 68.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

   9. associate-*r* 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   10. associate-*r* 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

   11. sqr-neg 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

   12. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   13. *-commutative 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   14. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   15. sqr-neg 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   16. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

3. Simplified 69.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   2. associate-*l* 77.5%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   3. associate-/r* 79.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   4. associate-*r* 83.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

   5. associate-*r* 90.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

   6. associate-/r* 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

   7. div-inv 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

   8. associate-/l/ 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   9. count-2 89.7%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   10. *-commutative 89.7%

       \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

5. Applied egg-rr 89.7%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
6. Step-by-step derivation

   1. associate-*r/ 89.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

   2. *-rgt-identity 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

   3. associate-/l/ 89.4%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

7. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

8. Taylor expanded in x around 0 58.1%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
9. Step-by-step derivation

   1. unpow2 58.1%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   2. associate-*r* 60.9%

      \[\leadsto \frac{1}{{c}^{2} \cdot \color{blue}{\left(\left({s}^{2} \cdot x\right) \cdot x\right)}} \]

   3. unpow2 60.9%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot x\right)} \]

   4. associate-*r* 64.5%

      \[\leadsto \frac{1}{{c}^{2} \cdot \left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot x\right)} \]

   5. associate-*l* 64.1%

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x}} \]

   6. unpow2 64.1%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x} \]

   7. associate-*r* 70.2%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)\right)} \cdot x} \]

   8. associate-*l* 70.7%

      \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right) \cdot x\right)}} \]

   9. *-commutative 70.7%

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot c\right)} \cdot x\right)} \]

   10. associate-*l* 75.7%

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(\left(s \cdot x\right) \cdot c\right)\right)} \cdot x\right)} \]

   11. *-commutative 75.7%

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right) \cdot x\right)} \]

   12. associate-*r* 75.7%

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right) \cdot x\right)} \]

   13. *-commutative 75.7%

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \cdot x\right)} \]

10. Simplified 75.7%

    \[\leadsto \color{blue}{\frac{1}{c \cdot \left(\left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right) \cdot x\right)}} \]

11. Final simplification 75.7%

    \[\leadsto \frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)} \]

Alternative 15: 38.1% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 4.1e+95) (/ 1.0 (* (- c c) (/ s s))) 0.0))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.1e+95) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 4.1d+95) then
        tmp = 1.0d0 / ((c - c) * (s / s))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.1e+95) {
		tmp = 1.0 / ((c - c) * (s / s));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 4.1e+95:
		tmp = 1.0 / ((c - c) * (s / s))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 4.1e+95)
		tmp = Float64(1.0 / Float64(Float64(c - c) * Float64(s / s)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 4.1e+95)
		tmp = 1.0 / ((c - c) * (s / s));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 4.1e+95], N[(1.0 / N[(N[(c - c), $MachinePrecision] * N[(s / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if s < 4.09999999999999986e95

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 70.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 57.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 57.4%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 60.7%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 60.8%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 60.7%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 60.7%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 63.6%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 35.3%

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\frac{s}{s}}} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \frac{1}{\color{blue}{\left(c - c\right)} \cdot \frac{s}{s}} \]

   if 4.09999999999999986e95 < s

   1. Initial program 64.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.8%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 64.1%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 61.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 65.5%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      2. associate-*l* 77.7%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      3. associate-/r* 77.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      4. associate-*r* 82.1%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      5. associate-*r* 88.8%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      6. associate-/r* 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

      7. div-inv 88.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

      8. associate-/l/ 88.7%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      9. count-2 88.7%

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      10. *-commutative 88.7%

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

      3. associate-/l/ 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

   7. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{c - c} \]
   10. Step-by-step derivation

       1. +-inverses 62.7%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 62.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\left(c - c\right) \cdot \frac{s}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 34.5% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 5.1e+95) (/ 1.0 (* c (* x c))) 0.0))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 5.1e+95) {
		tmp = 1.0 / (c * (x * c));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 5.1d+95) then
        tmp = 1.0d0 / (c * (x * c))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 5.1e+95) {
		tmp = 1.0 / (c * (x * c));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 5.1e+95:
		tmp = 1.0 / (c * (x * c))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 5.1e+95)
		tmp = Float64(1.0 / Float64(c * Float64(x * c)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 5.1e+95)
		tmp = 1.0 / (c * (x * c));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 5.1e+95], N[(1.0 / N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if s < 5.10000000000000003e95

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 70.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 57.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 57.4%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 60.7%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 60.8%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 60.7%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 60.7%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 63.6%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 35.3%

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\frac{s}{s}}} \]

   9. Taylor expanded in c around 0 35.3%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot x}} \]
   10. Step-by-step derivation

       1. unpow2 35.3%

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot x} \]

       2. associate-*l* 32.8%

          \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot x\right)}} \]

   11. Simplified 32.8%

       \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot x\right)}} \]

   if 5.10000000000000003e95 < s

   1. Initial program 64.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.8%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 64.1%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 61.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 65.5%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      2. associate-*l* 77.7%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      3. associate-/r* 77.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      4. associate-*r* 82.1%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      5. associate-*r* 88.8%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      6. associate-/r* 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

      7. div-inv 88.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

      8. associate-/l/ 88.7%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      9. count-2 88.7%

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      10. *-commutative 88.7%

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

      3. associate-/l/ 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

   7. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{c - c} \]
   10. Step-by-step derivation

       1. +-inverses 62.7%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 62.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 17: 36.0% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 5.4e+95) (/ (/ 1.0 x) (* c c)) 0.0))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 5.4e+95) {
		tmp = (1.0 / x) / (c * c);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 5.4d+95) then
        tmp = (1.0d0 / x) / (c * c)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 5.4e+95) {
		tmp = (1.0 / x) / (c * c);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 5.4e+95:
		tmp = (1.0 / x) / (c * c)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 5.4e+95)
		tmp = Float64(Float64(1.0 / x) / Float64(c * c));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 5.4e+95)
		tmp = (1.0 / x) / (c * c);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 5.4e+95], N[(N[(1.0 / x), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if s < 5.4e95

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 69.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 70.5%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 70.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 70.5%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.9%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 70.5%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 57.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 57.4%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      2. *-commutative 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

      3. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      4. unpow2 57.4%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      5. associate-*r* 60.7%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      6. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      7. *-commutative 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      8. associate-*r* 60.8%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      9. associate-*r* 60.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      10. *-commutative 60.7%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

      11. *-commutative 60.7%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

      12. associate-*r* 63.6%

          \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   6. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

   8. Applied egg-rr 35.3%

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\frac{s}{s}}} \]

   9. Taylor expanded in c around 0 35.3%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot x}} \]
   10. Step-by-step derivation

       1. associate-/l/ 35.3%

          \[\leadsto \color{blue}{\frac{\frac{1}{x}}{{c}^{2}}} \]

       2. unpow2 35.3%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{c \cdot c}} \]

   11. Simplified 35.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{c \cdot c}} \]

   if 5.4e95 < s

   1. Initial program 64.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.8%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. remove-double-neg 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      3. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

      4. distribute-lft-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

      5. distribute-rgt-neg-out 61.8%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

      6. associate-/r* 64.1%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

      7. *-commutative 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      8. associate-*r* 61.9%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

      9. associate-*r* 64.1%

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

      10. associate-*r* 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

      11. sqr-neg 63.8%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

      12. associate-*r* 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      13. *-commutative 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      14. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      15. sqr-neg 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      16. unpow2 64.1%

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 65.5%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      2. associate-*l* 77.7%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

      3. associate-/r* 77.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

      4. associate-*r* 82.1%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      5. associate-*r* 88.8%

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      6. associate-/r* 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

      7. div-inv 88.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

      8. associate-/l/ 88.7%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      9. count-2 88.7%

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

      10. *-commutative 88.7%

          \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

      2. *-rgt-identity 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

      3. associate-/l/ 88.8%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

   7. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{c - c} \]
   10. Step-by-step derivation

       1. +-inverses 62.7%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 62.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 18: 2.2% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 -0.5)

C

double code(double x, double c, double s) {
	return -0.5;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = -0.5d0
end function

Java

public static double code(double x, double c, double s) {
	return -0.5;
}

Python

def code(x, c, s):
	return -0.5

Julia

function code(x, c, s)
	return -0.5
end

MATLAB

function tmp = code(x, c, s)
	tmp = -0.5;
end

Wolfram

code[x_, c_, s_] := -0.5

Derivation

1. Initial program 69.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 68.3%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/r* 69.5%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

   7. *-commutative 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   8. associate-*r* 68.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

   9. associate-*r* 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   10. associate-*r* 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

   11. sqr-neg 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

   12. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   13. *-commutative 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   14. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   15. sqr-neg 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   16. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

3. Simplified 69.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

4. Taylor expanded in x around 0 58.1%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
5. Step-by-step derivation

   1. unpow2 58.1%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   2. *-commutative 58.1%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

   3. unpow2 58.1%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

   4. unpow2 58.1%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   5. associate-*r* 60.9%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   6. associate-*r* 60.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   7. *-commutative 60.5%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

   8. associate-*r* 61.0%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   9. associate-*r* 60.5%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   10. *-commutative 60.5%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(x \cdot \left(s \cdot s\right)\right)} \]

   11. *-commutative 60.5%

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

   12. associate-*r* 63.4%

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

6. Simplified 63.4%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}} \]

8. Applied egg-rr 34.5%

   \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\frac{s}{s}}} \]

10. Applied egg-rr 3.3%

    \[\leadsto \frac{1}{\color{blue}{\left(-2 - c\right)} \cdot \frac{s}{s}} \]

11. Taylor expanded in c around 0 2.2%

    \[\leadsto \color{blue}{-0.5} \]

12. Final simplification 2.2%

    \[\leadsto -0.5 \]

Alternative 19: 32.9% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 0.0)

C

double code(double x, double c, double s) {
	return 0.0;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 0.0d0
end function

Java

public static double code(double x, double c, double s) {
	return 0.0;
}

Python

def code(x, c, s):
	return 0.0

Julia

function code(x, c, s)
	return 0.0
end

MATLAB

function tmp = code(x, c, s)
	tmp = 0.0;
end

Wolfram

code[x_, c_, s_] := 0.0

Derivation

1. Initial program 69.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 68.3%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. remove-double-neg 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{-\left(-\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   3. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(-x \cdot {s}^{2}\right) \cdot x}} \]

   4. distribute-lft-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{-\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right)} \cdot x} \]

   5. distribute-rgt-neg-out 68.3%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)}} \]

   6. associate-/r* 69.5%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(\left(-x\right) \cdot {s}^{2}\right) \cdot \left(-x\right)\right)}} \]

   7. *-commutative 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   8. associate-*r* 68.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(-x\right)\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)}} \]

   9. associate-*r* 69.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left(\left(-x\right) \cdot \left(\left(-x\right) \cdot {s}^{2}\right)\right)}} \]

   10. associate-*r* 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot {s}^{2}\right)}} \]

   11. sqr-neg 63.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right)} \]

   12. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   13. *-commutative 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   14. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   15. sqr-neg 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(-c\right) \cdot \left(-c\right)\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   16. unpow2 69.5%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(-c\right)}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

3. Simplified 69.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 69.7%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   2. associate-*l* 77.5%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]

   3. associate-/r* 79.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot \left(s \cdot s\right)\right)}} \]

   4. associate-*r* 83.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

   5. associate-*r* 90.4%

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot s}} \]

   6. associate-/r* 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)}}{s}} \]

   7. div-inv 89.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s}} \]

   8. associate-/l/ 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   9. count-2 89.7%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x + x\right)}}{c \cdot x}}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{s} \]

   10. *-commutative 89.7%

       \[\leadsto \frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \color{blue}{\left(s \cdot x\right)}} \cdot \frac{1}{s} \]

5. Applied egg-rr 89.7%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{s}} \]
6. Step-by-step derivation

   1. associate-*r/ 89.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)} \cdot 1}{s}} \]

   2. *-rgt-identity 89.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot x}}{c \cdot \left(s \cdot x\right)}}}{s} \]

   3. associate-/l/ 89.4%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}}{s} \]

7. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)}}{s}} \]

9. Applied egg-rr 35.2%

   \[\leadsto \color{blue}{c - c} \]
10. Step-by-step derivation

    1. +-inverses 35.2%

       \[\leadsto \color{blue}{0} \]

11. Simplified 35.2%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 35.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))