mixedcos

Percentage Accurate: 67.0% → 97.4%

Time: 18.0s

Alternatives: 15

Speedup: 24.1×

• 32.1% 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: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = x * (c * s)
    code = (1.0d0 / t_0) * (cos((x * 2.0d0)) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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-/l/ 67.0%

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

   7. distribute-rgt-neg-out 67.0%

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

   8. distribute-lft-neg-out 67.0%

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

   9. associate-*l* 68.6%

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

   10. distribute-lft-neg-in 68.6%

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

   11. distribute-lft-neg-out 68.6%

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

   12. remove-double-neg 68.6%

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

   13. associate-*r* 68.7%

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

   14. *-commutative 68.7%

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

   15. associate-*r* 69.0%

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

3. Simplified 72.7%

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

4. Taylor expanded in c around 0 72.7%

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

   1. unpow2 72.7%

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

   2. associate-*r* 84.0%

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

6. Simplified 84.0%

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

   1. *-commutative 84.0%

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

   2. *-un-lft-identity 84.0%

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

   3. add-sqr-sqrt 83.9%

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

   4. times-frac 83.9%

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

8. Applied egg-rr 97.7%

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

9. Final simplification 97.7%

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

Alternative 2: 79.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+166}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 3.1e-5)
     (*
      (/ 1.0 (* c (* x s)))
      (* (/ 1.0 c) (+ (* -2.0 (/ x s)) (/ 1.0 (* x s)))))
     (if (<= x 2.6e+166)
       (/ t_0 (* x (* x (* c (* s (* c s))))))
       (/ t_0 (* x (* (* s s) (* c (* x c)))))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 3.1e-5) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else if (x <= 2.6e+166) {
		tmp = t_0 / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 3.1d-5) then
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) * (((-2.0d0) * (x / s)) + (1.0d0 / (x * s))))
    else if (x <= 2.6d+166) then
        tmp = t_0 / (x * (x * (c * (s * (c * s)))))
    else
        tmp = t_0 / (x * ((s * s) * (c * (x * c))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 3.1e-5) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else if (x <= 2.6e+166) {
		tmp = t_0 / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 3.1e-5:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))))
	elif x <= 2.6e+166:
		tmp = t_0 / (x * (x * (c * (s * (c * s)))))
	else:
		tmp = t_0 / (x * ((s * s) * (c * (x * c))))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 3.1e-5)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) * Float64(Float64(-2.0 * Float64(x / s)) + Float64(1.0 / Float64(x * s)))));
	elseif (x <= 2.6e+166)
		tmp = Float64(t_0 / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(s * s) * Float64(c * Float64(x * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 3.1e-5)
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	elseif (x <= 2.6e+166)
		tmp = t_0 / (x * (x * (c * (s * (c * s)))));
	else
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.1e-5], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(-2.0 * N[(x / s), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+166], N[(t$95$0 / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.10000000000000014e-5

   1. Initial program 66.0%

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

      1. associate-/r* 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-lft-neg-out 65.6%

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

      4. distribute-lft-neg-out 65.6%

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

      5. distribute-rgt-neg-out 65.6%

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

      6. associate-/r* 66.0%

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

      7. unpow2 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 63.2%

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

      10. sqr-neg 63.2%

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

      11. associate-*r* 66.0%

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

      12. unpow2 66.0%

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

   3. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*r* 63.2%

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

      3. associate-*l* 64.4%

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

      4. *-commutative 64.4%

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

      5. swap-sqr 80.4%

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

      6. add-sqr-sqrt 80.4%

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

      7. *-un-lft-identity 80.4%

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

      8. times-frac 80.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in x around 0 77.6%

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

   if 3.10000000000000014e-5 < x < 2.5999999999999999e166

   1. Initial program 84.1%

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

      1. associate-/r* 84.0%

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

      2. remove-double-neg 84.0%

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

      3. distribute-lft-neg-out 84.0%

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

      4. distribute-lft-neg-out 84.0%

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

      5. distribute-rgt-neg-out 84.0%

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

      6. associate-/l/ 84.1%

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

      7. distribute-rgt-neg-out 84.1%

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

      8. distribute-lft-neg-out 84.1%

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

      9. associate-*l* 84.0%

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

      10. distribute-lft-neg-in 84.0%

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

      11. distribute-lft-neg-out 84.0%

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

      12. remove-double-neg 84.0%

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

      13. associate-*r* 84.0%

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

      14. *-commutative 84.0%

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

      15. associate-*r* 84.1%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in c around 0 84.2%

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

      1. unpow2 84.2%

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

      2. associate-*r* 91.9%

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

   6. Simplified 91.9%

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

   if 2.5999999999999999e166 < x

   1. Initial program 55.9%

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

      1. associate-/r* 55.9%

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

      2. remove-double-neg 55.9%

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

      3. distribute-lft-neg-out 55.9%

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

      4. distribute-lft-neg-out 55.9%

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

      5. distribute-rgt-neg-out 55.9%

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

      6. associate-/l/ 55.9%

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

      7. distribute-rgt-neg-out 55.9%

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

      8. distribute-lft-neg-out 55.9%

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

      9. associate-*l* 59.1%

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

      10. distribute-lft-neg-in 59.1%

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

      11. distribute-lft-neg-out 59.1%

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

      12. remove-double-neg 59.1%

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

      13. associate-*r* 59.2%

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

      14. *-commutative 59.2%

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

      15. associate-*r* 59.4%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in x around 0 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*r* 62.5%

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

   6. Simplified 62.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+166}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 84.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 1.05 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{t_0}{c \cdot s}}{s \cdot \left(c \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 1.05e-25)
     (* (/ 1.0 (* c (* x s))) (* (/ 1.0 c) (/ 1.0 (* x s))))
     (if (<= x 1.35e+154)
       (/ (/ t_0 (* c s)) (* s (* c (* x x))))
       (/ t_0 (* x (* (* s s) (* c (* x c)))))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 1.05e-25) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)));
	} else if (x <= 1.35e+154) {
		tmp = (t_0 / (c * s)) / (s * (c * (x * x)));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 1.05d-25) then
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) * (1.0d0 / (x * s)))
    else if (x <= 1.35d+154) then
        tmp = (t_0 / (c * s)) / (s * (c * (x * x)))
    else
        tmp = t_0 / (x * ((s * s) * (c * (x * c))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 1.05e-25) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)));
	} else if (x <= 1.35e+154) {
		tmp = (t_0 / (c * s)) / (s * (c * (x * x)));
	} else {
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 1.05e-25:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)))
	elif x <= 1.35e+154:
		tmp = (t_0 / (c * s)) / (s * (c * (x * x)))
	else:
		tmp = t_0 / (x * ((s * s) * (c * (x * c))))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 1.05e-25)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) * Float64(1.0 / Float64(x * s))));
	elseif (x <= 1.35e+154)
		tmp = Float64(Float64(t_0 / Float64(c * s)) / Float64(s * Float64(c * Float64(x * x))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(s * s) * Float64(c * Float64(x * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 1.05e-25)
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)));
	elseif (x <= 1.35e+154)
		tmp = (t_0 / (c * s)) / (s * (c * (x * x)));
	else
		tmp = t_0 / (x * ((s * s) * (c * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.05e-25], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[(t$95$0 / N[(c * s), $MachinePrecision]), $MachinePrecision] / N[(s * N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(s * s), $MachinePrecision] * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.05000000000000001e-25

   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* 65.3%

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

      2. remove-double-neg 65.3%

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

      3. distribute-lft-neg-out 65.3%

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

      4. distribute-lft-neg-out 65.3%

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

      5. distribute-rgt-neg-out 65.3%

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

      6. associate-/r* 65.8%

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

      7. unpow2 65.8%

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

      8. *-commutative 65.8%

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

      9. associate-*r* 63.0%

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

      10. sqr-neg 63.0%

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

      11. associate-*r* 65.8%

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

      12. unpow2 65.8%

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

   3. Simplified 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r* 63.0%

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

      3. associate-*l* 64.6%

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

      4. *-commutative 64.6%

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

      5. swap-sqr 81.0%

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

      6. add-sqr-sqrt 80.9%

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

      7. *-un-lft-identity 80.9%

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

      8. times-frac 80.9%

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

   5. Applied egg-rr 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 98.2%

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in x around 0 86.0%

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

      1. *-commutative 86.0%

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

   10. Simplified 86.0%

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

   if 1.05000000000000001e-25 < x < 1.35000000000000003e154

   1. Initial program 82.7%

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

      1. associate-/r* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-lft-neg-out 82.7%

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

      4. distribute-lft-neg-out 82.7%

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

      5. distribute-rgt-neg-out 82.7%

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

      6. associate-/r* 82.7%

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

      7. unpow2 82.7%

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

      8. *-commutative 82.7%

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

      9. associate-*r* 82.8%

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

      10. sqr-neg 82.8%

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

      11. associate-*r* 82.7%

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

      12. unpow2 82.7%

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

   3. Simplified 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. associate-*r* 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-*l* 79.9%

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

      5. swap-sqr 90.3%

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

      6. *-commutative 90.3%

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

      7. associate-*r* 98.5%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. associate-*r* 93.7%

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

   7. Applied egg-rr 93.7%

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

   if 1.35000000000000003e154 < x

   1. Initial program 57.4%

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

      1. associate-/r* 57.4%

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

      2. remove-double-neg 57.4%

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

      3. distribute-lft-neg-out 57.4%

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

      4. distribute-lft-neg-out 57.4%

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

      5. distribute-rgt-neg-out 57.4%

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

      6. associate-/l/ 57.4%

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

      7. distribute-rgt-neg-out 57.4%

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

      8. distribute-lft-neg-out 57.4%

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

      9. associate-*l* 60.5%

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

      10. distribute-lft-neg-in 60.5%

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

      11. distribute-lft-neg-out 60.5%

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

      12. remove-double-neg 60.5%

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

      13. associate-*r* 60.5%

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

      14. *-commutative 60.5%

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

      15. associate-*r* 60.8%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. unpow2 60.5%

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

      2. unpow2 60.5%

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

      3. associate-*r* 63.8%

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

   6. Simplified 63.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot s}}{s \cdot \left(c \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 4: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 3.1e-5)
   (* (/ 1.0 (* c (* x s))) (* (/ 1.0 c) (+ (* -2.0 (/ x s)) (/ 1.0 (* x s)))))
   (/ (cos (* x 2.0)) (* x (* x (* c (* c (* s s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.1e-5) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 3.1d-5) then
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) * (((-2.0d0) * (x / s)) + (1.0d0 / (x * s))))
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (c * (s * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.1e-5) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 3.1e-5:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))))
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 3.1e-5)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) * Float64(Float64(-2.0 * Float64(x / s)) + Float64(1.0 / Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 3.1e-5)
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 3.1e-5], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(-2.0 * N[(x / s), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000014e-5

   1. Initial program 66.0%

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

      1. associate-/r* 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-lft-neg-out 65.6%

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

      4. distribute-lft-neg-out 65.6%

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

      5. distribute-rgt-neg-out 65.6%

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

      6. associate-/r* 66.0%

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

      7. unpow2 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 63.2%

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

      10. sqr-neg 63.2%

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

      11. associate-*r* 66.0%

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

      12. unpow2 66.0%

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

   3. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*r* 63.2%

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

      3. associate-*l* 64.4%

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

      4. *-commutative 64.4%

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

      5. swap-sqr 80.4%

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

      6. add-sqr-sqrt 80.4%

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

      7. *-un-lft-identity 80.4%

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

      8. times-frac 80.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in x around 0 77.6%

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

   if 3.10000000000000014e-5 < x

   1. Initial program 70.3%

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

      1. associate-/r* 70.2%

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

      2. remove-double-neg 70.2%

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

      3. distribute-lft-neg-out 70.2%

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

      4. distribute-lft-neg-out 70.2%

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

      5. distribute-rgt-neg-out 70.2%

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

      6. associate-/l/ 70.3%

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

      7. distribute-rgt-neg-out 70.3%

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

      8. distribute-lft-neg-out 70.3%

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

      9. associate-*l* 71.8%

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

      10. distribute-lft-neg-in 71.8%

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

      11. distribute-lft-neg-out 71.8%

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

      12. remove-double-neg 71.8%

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

      13. associate-*r* 71.8%

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

      14. *-commutative 71.8%

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

      15. associate-*r* 72.0%

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

   3. Simplified 73.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 79.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 6.8e-6)
   (* (/ 1.0 (* c (* x s))) (* (/ 1.0 c) (+ (* -2.0 (/ x s)) (/ 1.0 (* x s)))))
   (/ (cos (* x 2.0)) (* x (* x (* c (* s (* c s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 6.8e-6) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (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 <= 6.8d-6) then
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) * (((-2.0d0) * (x / s)) + (1.0d0 / (x * s))))
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (s * (c * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 6.8e-6) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 6.8e-6:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))))
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 6.8e-6)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) * Float64(Float64(-2.0 * Float64(x / s)) + Float64(1.0 / Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 6.8e-6)
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * ((-2.0 * (x / s)) + (1.0 / (x * s))));
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 6.8e-6], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(N[(-2.0 * N[(x / s), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.80000000000000012e-6

   1. Initial program 66.0%

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

      1. associate-/r* 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-lft-neg-out 65.6%

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

      4. distribute-lft-neg-out 65.6%

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

      5. distribute-rgt-neg-out 65.6%

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

      6. associate-/r* 66.0%

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

      7. unpow2 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 63.2%

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

      10. sqr-neg 63.2%

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

      11. associate-*r* 66.0%

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

      12. unpow2 66.0%

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

   3. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*r* 63.2%

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

      3. associate-*l* 64.4%

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

      4. *-commutative 64.4%

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

      5. swap-sqr 80.4%

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

      6. add-sqr-sqrt 80.4%

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

      7. *-un-lft-identity 80.4%

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

      8. times-frac 80.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in x around 0 77.6%

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

   if 6.80000000000000012e-6 < x

   1. Initial program 70.3%

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

      1. associate-/r* 70.2%

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

      2. remove-double-neg 70.2%

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

      3. distribute-lft-neg-out 70.2%

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

      4. distribute-lft-neg-out 70.2%

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

      5. distribute-rgt-neg-out 70.2%

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

      6. associate-/l/ 70.3%

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

      7. distribute-rgt-neg-out 70.3%

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

      8. distribute-lft-neg-out 70.3%

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

      9. associate-*l* 71.8%

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

      10. distribute-lft-neg-in 71.8%

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

      11. distribute-lft-neg-out 71.8%

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

      12. remove-double-neg 71.8%

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

      13. associate-*r* 71.8%

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

      14. *-commutative 71.8%

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

      15. associate-*r* 72.0%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in c around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*r* 82.7%

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

   6. Simplified 82.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \left(-2 \cdot \frac{x}{s} + \frac{1}{x \cdot s}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = (1.0d0 / t_0) * (cos((x * 2.0d0)) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*r* 64.1%

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

   3. associate-*l* 65.0%

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

   4. *-commutative 65.0%

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

   5. swap-sqr 79.8%

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

   6. add-sqr-sqrt 79.7%

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

   7. *-un-lft-identity 79.7%

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

   8. times-frac 79.7%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 7: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}}{s \cdot s}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 2.6e-177)
   (* (/ 1.0 c) (/ (- (/ 1.0 (* c (* x x))) (/ 2.0 c)) (* s s)))
   (pow (* c (* x s)) -2.0)))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 2.6e-177) {
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 2.6d-177) then
        tmp = (1.0d0 / c) * (((1.0d0 / (c * (x * x))) - (2.0d0 / c)) / (s * s))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 2.6e-177) {
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 2.6e-177:
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 2.6e-177)
		tmp = Float64(Float64(1.0 / c) * Float64(Float64(Float64(1.0 / Float64(c * Float64(x * x))) - Float64(2.0 / c)) / Float64(s * s)));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 2.6e-177)
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 2.6e-177], N[(N[(1.0 / c), $MachinePrecision] * N[(N[(N[(1.0 / N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision] / N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 2.6000000000000001e-177

   1. Initial program 64.8%

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

      1. associate-/r* 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* 64.8%

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

      7. unpow2 64.8%

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

      8. *-commutative 64.8%

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

      9. associate-*r* 60.8%

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

      10. sqr-neg 60.8%

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

      11. associate-*r* 64.8%

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

      12. unpow2 64.8%

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

   3. Simplified 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. associate-*l* 69.0%

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

      3. times-frac 69.4%

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

      4. *-commutative 69.4%

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

      5. associate-*r* 64.8%

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

      6. pow2 64.8%

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

      7. pow2 64.8%

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

      8. pow-prod-down 84.6%

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

      9. *-commutative 84.6%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in x around 0 30.8%

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

      1. associate-/r* 30.6%

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

      2. unpow2 30.6%

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

      3. unpow2 30.6%

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

      4. associate-*r/ 30.6%

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

      5. metadata-eval 30.6%

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

      6. unpow2 30.6%

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

      7. associate-*l* 33.3%

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

      8. *-commutative 33.3%

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

      9. *-commutative 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in s around 0 53.4%

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

       1. unpow2 53.4%

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

       2. associate-*r/ 53.4%

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

       3. metadata-eval 53.4%

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

       4. unpow2 53.4%

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

   11. Simplified 53.4%

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

   if 2.6000000000000001e-177 < s

   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* 70.4%

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

      2. remove-double-neg 70.4%

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

      3. distribute-lft-neg-out 70.4%

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

      4. distribute-lft-neg-out 70.4%

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

      5. distribute-rgt-neg-out 70.4%

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

      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. unpow2 70.5%

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

      8. *-commutative 70.5%

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

      9. associate-*r* 69.2%

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

      10. sqr-neg 69.2%

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

      11. associate-*r* 70.5%

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

      12. unpow2 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 63.0%

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

      1. unpow2 63.0%

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

      2. associate-/r* 62.9%

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

      3. unpow2 62.9%

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

      4. unpow2 62.9%

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

      5. swap-sqr 78.0%

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

      6. unpow2 78.0%

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

      7. associate-/r* 77.9%

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

      8. unpow2 77.9%

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

      9. swap-sqr 90.9%

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

      10. unpow2 90.9%

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

      11. associate-*r* 90.0%

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

      12. *-commutative 90.0%

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

      13. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. pow-flip 87.7%

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

      2. associate-*r* 90.1%

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

      3. *-commutative 90.1%

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

      4. associate-*r* 91.0%

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

      5. metadata-eval 91.0%

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

   8. Applied egg-rr 91.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}}{s \cdot s}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 8: 65.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}}{s \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.8e-179)
   (* (/ 1.0 c) (/ (- (/ 1.0 (* c (* x x))) (/ 2.0 c)) (* s s)))
   (* (/ 1.0 (* c (* x s))) (* (/ 1.0 c) (/ 1.0 (* x s))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.8e-179) {
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	} else {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (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 (s <= 1.8d-179) then
        tmp = (1.0d0 / c) * (((1.0d0 / (c * (x * x))) - (2.0d0 / c)) / (s * s))
    else
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) * (1.0d0 / (x * s)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.8e-179) {
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	} else {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.8e-179:
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s))
	else:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.8e-179)
		tmp = Float64(Float64(1.0 / c) * Float64(Float64(Float64(1.0 / Float64(c * Float64(x * x))) - Float64(2.0 / c)) / Float64(s * s)));
	else
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) * Float64(1.0 / Float64(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 1.8e-179)
		tmp = (1.0 / c) * (((1.0 / (c * (x * x))) - (2.0 / c)) / (s * s));
	else
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) * (1.0 / (x * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.8e-179], N[(N[(1.0 / c), $MachinePrecision] * N[(N[(N[(1.0 / N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision] / N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] * N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.80000000000000004e-179

   1. Initial program 64.8%

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

      1. associate-/r* 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* 64.8%

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

      7. unpow2 64.8%

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

      8. *-commutative 64.8%

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

      9. associate-*r* 60.8%

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

      10. sqr-neg 60.8%

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

      11. associate-*r* 64.8%

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

      12. unpow2 64.8%

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

   3. Simplified 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. associate-*l* 69.0%

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

      3. times-frac 69.4%

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

      4. *-commutative 69.4%

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

      5. associate-*r* 64.8%

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

      6. pow2 64.8%

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

      7. pow2 64.8%

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

      8. pow-prod-down 84.6%

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

      9. *-commutative 84.6%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in x around 0 30.8%

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

      1. associate-/r* 30.6%

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

      2. unpow2 30.6%

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

      3. unpow2 30.6%

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

      4. associate-*r/ 30.6%

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

      5. metadata-eval 30.6%

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

      6. unpow2 30.6%

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

      7. associate-*l* 33.3%

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

      8. *-commutative 33.3%

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

      9. *-commutative 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in s around 0 53.4%

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

       1. unpow2 53.4%

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

       2. associate-*r/ 53.4%

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

       3. metadata-eval 53.4%

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

       4. unpow2 53.4%

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

   11. Simplified 53.4%

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

   if 1.80000000000000004e-179 < s

   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* 70.4%

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

      2. remove-double-neg 70.4%

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

      3. distribute-lft-neg-out 70.4%

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

      4. distribute-lft-neg-out 70.4%

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

      5. distribute-rgt-neg-out 70.4%

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

      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. unpow2 70.5%

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

      8. *-commutative 70.5%

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

      9. associate-*r* 69.2%

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

      10. sqr-neg 69.2%

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

      11. associate-*r* 70.5%

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

      12. unpow2 70.5%

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

   3. Simplified 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*r* 69.2%

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

      3. associate-*l* 71.3%

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

      4. *-commutative 71.3%

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

      5. swap-sqr 85.8%

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

      6. add-sqr-sqrt 85.8%

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

      7. *-un-lft-identity 85.8%

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

      8. times-frac 85.7%

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

   5. Applied egg-rr 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. times-frac 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around 0 91.0%

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

      1. *-commutative 91.0%

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

   10. Simplified 91.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot x\right)} - \frac{2}{c}}{s \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \end{array} \]

Alternative 9: 78.7% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*r* 64.1%

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

   3. associate-*l* 65.0%

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

   4. *-commutative 65.0%

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

   5. swap-sqr 79.8%

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

   6. add-sqr-sqrt 79.7%

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

   7. *-un-lft-identity 79.7%

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

   8. times-frac 79.7%

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

5. Applied egg-rr 98.0%

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

   1. *-un-lft-identity 98.0%

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

   2. times-frac 97.8%

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

7. Applied egg-rr 97.8%

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

8. Taylor expanded in x around 0 80.6%

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

   1. *-commutative 80.6%

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

10. Simplified 80.6%

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

11. Final simplification 80.6%

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

Alternative 10: 78.8% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*r* 64.1%

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

   3. associate-*l* 65.0%

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

   4. *-commutative 65.0%

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

   5. swap-sqr 79.8%

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

   6. add-sqr-sqrt 79.7%

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

   7. *-un-lft-identity 79.7%

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

   8. times-frac 79.7%

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

5. Applied egg-rr 98.0%

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

6. Taylor expanded in x around 0 80.8%

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

7. Final simplification 80.8%

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

Alternative 11: 64.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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-/l/ 67.0%

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

   7. distribute-rgt-neg-out 67.0%

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

   8. distribute-lft-neg-out 67.0%

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

   9. associate-*l* 68.6%

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

   10. distribute-lft-neg-in 68.6%

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

   11. distribute-lft-neg-out 68.6%

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

   12. remove-double-neg 68.6%

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

   13. associate-*r* 68.7%

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

   14. *-commutative 68.7%

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

   15. associate-*r* 69.0%

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

3. Simplified 72.7%

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

4. Taylor expanded in x around 0 63.5%

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

5. Final simplification 63.5%

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

Alternative 12: 70.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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-/l/ 67.0%

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

   7. distribute-rgt-neg-out 67.0%

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

   8. distribute-lft-neg-out 67.0%

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

   9. associate-*l* 68.6%

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

   10. distribute-lft-neg-in 68.6%

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

   11. distribute-lft-neg-out 68.6%

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

   12. remove-double-neg 68.6%

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

   13. associate-*r* 68.7%

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

   14. *-commutative 68.7%

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

   15. associate-*r* 69.0%

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

3. Simplified 72.7%

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

4. Taylor expanded in x around 0 63.5%

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

5. Taylor expanded in c around 0 63.5%

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

   1. unpow2 72.7%

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

   2. associate-*r* 84.0%

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

7. Simplified 71.9%

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

8. Final simplification 71.9%

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

Alternative 13: 78.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

4. Taylor expanded in x around 0 56.6%

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

   1. unpow2 56.6%

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

   2. associate-/r* 55.9%

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

   3. unpow2 55.9%

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

   4. unpow2 55.9%

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

   5. swap-sqr 67.7%

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

   6. unpow2 67.7%

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

   7. associate-/r* 68.4%

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

   8. unpow2 68.4%

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

   9. swap-sqr 80.5%

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

   10. unpow2 80.5%

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

   11. associate-*r* 79.6%

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

   12. *-commutative 79.6%

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

   13. associate-*l* 79.1%

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

6. Simplified 79.1%

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

   1. associate-*r* 79.6%

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

   2. *-commutative 79.6%

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

   3. associate-*r* 80.5%

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

   4. pow2 80.5%

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

8. Applied egg-rr 80.5%

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

9. Final simplification 80.5%

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

Alternative 14: 27.9% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c} \cdot \frac{\frac{-2}{c}}{s \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, c, s):
	return (1.0 / c) * ((-2.0 / c) / (s * s))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

   1. *-un-lft-identity 67.0%

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

   2. associate-*l* 70.8%

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

   3. times-frac 71.1%

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

   4. *-commutative 71.1%

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

   5. associate-*r* 67.8%

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

   6. pow2 67.8%

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

   7. pow2 67.8%

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

   8. pow-prod-down 85.8%

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

   9. *-commutative 85.8%

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

5. Applied egg-rr 85.8%

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

6. Taylor expanded in x around 0 39.5%

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

   1. associate-/r* 39.4%

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

   2. unpow2 39.4%

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

   3. unpow2 39.4%

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

   4. associate-*r/ 39.4%

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

   5. metadata-eval 39.4%

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

   6. unpow2 39.4%

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

   7. associate-*l* 40.6%

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

   8. *-commutative 40.6%

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

   9. *-commutative 40.6%

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

8. Simplified 40.6%

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

9. Taylor expanded in x around inf 28.0%

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

    1. associate-/r* 28.0%

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

    2. unpow2 28.0%

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

11. Simplified 28.0%

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

12. Final simplification 28.0%

    \[\leadsto \frac{1}{c} \cdot \frac{\frac{-2}{c}}{s \cdot s} \]

Alternative 15: 27.7% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-2}{s \cdot s}}{c \cdot c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 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* 67.0%

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

   7. unpow2 67.0%

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

   8. *-commutative 67.0%

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

   9. associate-*r* 64.1%

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

   10. sqr-neg 64.1%

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

   11. associate-*r* 67.0%

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

   12. unpow2 67.0%

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

3. Simplified 67.0%

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

   1. *-un-lft-identity 67.0%

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

   2. associate-*l* 70.8%

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

   3. times-frac 71.1%

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

   4. *-commutative 71.1%

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

   5. associate-*r* 67.8%

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

   6. pow2 67.8%

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

   7. pow2 67.8%

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

   8. pow-prod-down 85.8%

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

   9. *-commutative 85.8%

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

5. Applied egg-rr 85.8%

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

6. Taylor expanded in x around 0 39.5%

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

   1. associate-/r* 39.4%

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

   2. unpow2 39.4%

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

   3. unpow2 39.4%

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

   4. associate-*r/ 39.4%

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

   5. metadata-eval 39.4%

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

   6. unpow2 39.4%

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

   7. associate-*l* 40.6%

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

   8. *-commutative 40.6%

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

   9. *-commutative 40.6%

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

8. Simplified 40.6%

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

9. Taylor expanded in x around inf 28.4%

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

    1. associate-/r* 28.4%

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

    2. unpow2 28.4%

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

    3. unpow2 28.4%

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

11. Simplified 28.4%

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

12. Final simplification 28.4%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))