mixedcos

Percentage Accurate: 67.0% → 97.9%

Time: 20.6s

Alternatives: 13

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 1e+120)
   (/ (cos (* x -2.0)) (pow (* c_m (* s_m x)) 2.0))
   (* (/ -1.0 (* x (- (* s_m c_m)))) (/ (/ (cos (* 2.0 x)) x) (* s_m c_m)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if (pow(s_m, 2.0) <= 1e+120) {
		tmp = cos((x * -2.0)) / pow((c_m * (s_m * x)), 2.0);
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((cos((2.0 * x)) / x) / (s_m * c_m));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((s_m ** 2.0d0) <= 1d+120) then
        tmp = cos((x * (-2.0d0))) / ((c_m * (s_m * x)) ** 2.0d0)
    else
        tmp = ((-1.0d0) / (x * -(s_m * c_m))) * ((cos((2.0d0 * x)) / x) / (s_m * c_m))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if (Math.pow(s_m, 2.0) <= 1e+120) {
		tmp = Math.cos((x * -2.0)) / Math.pow((c_m * (s_m * x)), 2.0);
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((Math.cos((2.0 * x)) / x) / (s_m * c_m));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if math.pow(s_m, 2.0) <= 1e+120:
		tmp = math.cos((x * -2.0)) / math.pow((c_m * (s_m * x)), 2.0)
	else:
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((math.cos((2.0 * x)) / x) / (s_m * c_m))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if ((s_m ^ 2.0) <= 1e+120)
		tmp = Float64(cos(Float64(x * -2.0)) / (Float64(c_m * Float64(s_m * x)) ^ 2.0));
	else
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-Float64(s_m * c_m)))) * Float64(Float64(cos(Float64(2.0 * x)) / x) / Float64(s_m * c_m)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 1e+120)
		tmp = cos((x * -2.0)) / ((c_m * (s_m * x)) ^ 2.0);
	else
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((cos((2.0 * x)) / x) / (s_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 1e+120], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x * (-N[(s$95$m * c$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 9.9999999999999998e119

   1. Initial program 68.2%

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

      1. associate-/r* 68.3%

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

      2. unpow2 68.3%

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

      3. sqr-neg 68.3%

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

      4. unpow2 68.3%

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

      5. associate-/r* 68.2%

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

      6. cos-neg 68.2%

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

      7. *-commutative 68.2%

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

      8. distribute-rgt-neg-in 68.2%

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

      9. metadata-eval 68.2%

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

      10. associate-*r* 70.3%

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

      11. *-commutative 70.3%

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

      12. unpow2 70.3%

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

      13. sqr-neg 70.3%

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

      14. associate-*l* 75.3%

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

      15. associate-*r* 77.1%

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

      16. associate-*r* 74.7%

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

      17. associate-*r* 68.5%

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

      18. unpow2 68.5%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. associate-/r* 62.0%

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

      2. *-commutative 62.0%

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

      3. unpow2 62.0%

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

      4. unpow2 62.0%

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

      5. swap-sqr 78.1%

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

      6. unpow2 78.1%

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

      7. associate-/r* 77.8%

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

      8. *-commutative 77.8%

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

      9. unpow2 77.8%

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

      10. unpow2 77.8%

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

      11. swap-sqr 98.3%

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

      12. unpow2 98.3%

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

      13. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if 9.9999999999999998e119 < (pow.f64 s 2)

   1. Initial program 60.9%

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

      1. *-un-lft-identity 60.9%

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

      2. add-sqr-sqrt 60.9%

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

      3. times-frac 60.9%

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

   4. Applied egg-rr 94.5%

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

      1. frac-2neg 94.5%

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

      2. metadata-eval 94.5%

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

      3. div-inv 94.5%

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

      4. *-commutative 94.5%

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

      5. distribute-rgt-neg-in 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. associate-*r/ 94.5%

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

      2. metadata-eval 94.5%

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

      3. associate-*l* 93.2%

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

   8. Simplified 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r* 98.3%

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

      4. times-frac 98.3%

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

      5. *-commutative 98.3%

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

   10. Applied egg-rr 98.3%

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

       1. associate-*l/ 98.3%

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

       2. *-lft-identity 98.3%

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

   12. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 2: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;{s_m}^{2} \leq 10^{+76}:\\ \;\;\;\;\frac{1}{\left(c_m \cdot \left(s_m \cdot x\right)\right) \cdot \frac{c_m}{\frac{t_0}{s_m \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-s_m \cdot c_m\right)} \cdot \frac{\frac{t_0}{s_m}}{x \cdot c_m}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (if (<= (pow s_m 2.0) 1e+76)
     (/ 1.0 (* (* c_m (* s_m x)) (/ c_m (/ t_0 (* s_m x)))))
     (* (/ -1.0 (* x (- (* s_m c_m)))) (/ (/ t_0 s_m) (* x c_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (pow(s_m, 2.0) <= 1e+76) {
		tmp = 1.0 / ((c_m * (s_m * x)) * (c_m / (t_0 / (s_m * x))));
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / s_m) / (x * c_m));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if ((s_m ** 2.0d0) <= 1d+76) then
        tmp = 1.0d0 / ((c_m * (s_m * x)) * (c_m / (t_0 / (s_m * x))))
    else
        tmp = ((-1.0d0) / (x * -(s_m * c_m))) * ((t_0 / s_m) / (x * c_m))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double tmp;
	if (Math.pow(s_m, 2.0) <= 1e+76) {
		tmp = 1.0 / ((c_m * (s_m * x)) * (c_m / (t_0 / (s_m * x))));
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / s_m) / (x * c_m));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if math.pow(s_m, 2.0) <= 1e+76:
		tmp = 1.0 / ((c_m * (s_m * x)) * (c_m / (t_0 / (s_m * x))))
	else:
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / s_m) / (x * c_m))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if ((s_m ^ 2.0) <= 1e+76)
		tmp = Float64(1.0 / Float64(Float64(c_m * Float64(s_m * x)) * Float64(c_m / Float64(t_0 / Float64(s_m * x)))));
	else
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-Float64(s_m * c_m)))) * Float64(Float64(t_0 / s_m) / Float64(x * c_m)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 1e+76)
		tmp = 1.0 / ((c_m * (s_m * x)) * (c_m / (t_0 / (s_m * x))));
	else
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / s_m) / (x * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 1e+76], N[(1.0 / N[(N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * N[(c$95$m / N[(t$95$0 / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x * (-N[(s$95$m * c$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 1e76

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.5%

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

   4. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. clear-num 98.5%

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

      3. frac-times 98.2%

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

      4. metadata-eval 98.2%

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

      5. associate-/l* 98.0%

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

      6. *-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

   if 1e76 < (pow.f64 s 2)

   1. Initial program 62.1%

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

      1. *-un-lft-identity 62.1%

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

      2. add-sqr-sqrt 62.1%

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

      3. times-frac 62.0%

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

   4. Applied egg-rr 94.7%

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

      1. frac-2neg 94.7%

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

      2. metadata-eval 94.7%

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

      3. div-inv 94.7%

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

      4. *-commutative 94.7%

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

      5. distribute-rgt-neg-in 94.7%

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

   6. Applied egg-rr 94.7%

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

      1. associate-*r/ 94.7%

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

      2. metadata-eval 94.7%

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

      3. associate-*l* 93.5%

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

   8. Simplified 93.5%

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

      1. *-un-lft-identity 93.5%

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

      2. associate-*r* 96.7%

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

      3. times-frac 96.7%

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

      4. *-commutative 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. associate-*l/ 96.8%

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

       2. *-lft-identity 96.8%

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

   12. Simplified 96.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 10^{+76}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \frac{c}{\frac{\cos \left(2 \cdot x\right)}{s \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-s \cdot c\right)} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{s}}{x \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := c_m \cdot \left(s_m \cdot x\right)\\ \mathbf{if}\;{s_m}^{2} \leq 10^{+120}:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-s_m \cdot c_m\right)} \cdot \frac{\frac{t_0}{x}}{s_m \cdot c_m}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (* c_m (* s_m x))))
   (if (<= (pow s_m 2.0) 1e+120)
     (/ (/ t_0 t_1) t_1)
     (* (/ -1.0 (* x (- (* s_m c_m)))) (/ (/ t_0 x) (* s_m c_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double t_1 = c_m * (s_m * x);
	double tmp;
	if (pow(s_m, 2.0) <= 1e+120) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / x) / (s_m * c_m));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = c_m * (s_m * x)
    if ((s_m ** 2.0d0) <= 1d+120) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = ((-1.0d0) / (x * -(s_m * c_m))) * ((t_0 / x) / (s_m * c_m))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double t_1 = c_m * (s_m * x);
	double tmp;
	if (Math.pow(s_m, 2.0) <= 1e+120) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / x) / (s_m * c_m));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	t_1 = c_m * (s_m * x)
	tmp = 0
	if math.pow(s_m, 2.0) <= 1e+120:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / x) / (s_m * c_m))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(c_m * Float64(s_m * x))
	tmp = 0.0
	if ((s_m ^ 2.0) <= 1e+120)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-Float64(s_m * c_m)))) * Float64(Float64(t_0 / x) / Float64(s_m * c_m)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((2.0 * x));
	t_1 = c_m * (s_m * x);
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 1e+120)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = (-1.0 / (x * -(s_m * c_m))) * ((t_0 / x) / (s_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 1e+120], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(-1.0 / N[(x * (-N[(s$95$m * c$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / x), $MachinePrecision] / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 9.9999999999999998e119

   1. Initial program 68.2%

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

      1. *-un-lft-identity 68.2%

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

      2. add-sqr-sqrt 68.2%

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

      3. times-frac 68.2%

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

   4. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l/ 98.5%

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

      3. div-inv 98.6%

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

      4. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 9.9999999999999998e119 < (pow.f64 s 2)

   1. Initial program 60.9%

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

      1. *-un-lft-identity 60.9%

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

      2. add-sqr-sqrt 60.9%

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

      3. times-frac 60.9%

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

   4. Applied egg-rr 94.5%

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

      1. frac-2neg 94.5%

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

      2. metadata-eval 94.5%

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

      3. div-inv 94.5%

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

      4. *-commutative 94.5%

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

      5. distribute-rgt-neg-in 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. associate-*r/ 94.5%

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

      2. metadata-eval 94.5%

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

      3. associate-*l* 93.2%

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

   8. Simplified 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r* 98.3%

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

      4. times-frac 98.3%

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

      5. *-commutative 98.3%

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

   10. Applied egg-rr 98.3%

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

       1. associate-*l/ 98.3%

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

       2. *-lft-identity 98.3%

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

   12. Simplified 98.3%

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

4. Final simplification 98.5%

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

Alternative 4: 95.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(s_m \cdot x\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;c_m \leq 8 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{t_1}{c_m}}{\left(s_m \cdot x\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{c_m \cdot t_0}}{s_m \cdot x}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x))) (t_1 (cos (* 2.0 x))))
   (if (<= c_m 8e-198)
     (/ (/ t_1 c_m) (* (* s_m x) t_0))
     (/ (/ t_1 (* c_m t_0)) (* s_m x)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double t_1 = cos((2.0 * x));
	double tmp;
	if (c_m <= 8e-198) {
		tmp = (t_1 / c_m) / ((s_m * x) * t_0);
	} else {
		tmp = (t_1 / (c_m * t_0)) / (s_m * x);
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (s_m * x)
    t_1 = cos((2.0d0 * x))
    if (c_m <= 8d-198) then
        tmp = (t_1 / c_m) / ((s_m * x) * t_0)
    else
        tmp = (t_1 / (c_m * t_0)) / (s_m * x)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if (c_m <= 8e-198) {
		tmp = (t_1 / c_m) / ((s_m * x) * t_0);
	} else {
		tmp = (t_1 / (c_m * t_0)) / (s_m * x);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if c_m <= 8e-198:
		tmp = (t_1 / c_m) / ((s_m * x) * t_0)
	else:
		tmp = (t_1 / (c_m * t_0)) / (s_m * x)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (c_m <= 8e-198)
		tmp = Float64(Float64(t_1 / c_m) / Float64(Float64(s_m * x) * t_0));
	else
		tmp = Float64(Float64(t_1 / Float64(c_m * t_0)) / Float64(s_m * x));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if (c_m <= 8e-198)
		tmp = (t_1 / c_m) / ((s_m * x) * t_0);
	else
		tmp = (t_1 / (c_m * t_0)) / (s_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 8e-198], N[(N[(t$95$1 / c$95$m), $MachinePrecision] / N[(N[(s$95$m * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(c$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 7.9999999999999993e-198

   1. Initial program 60.7%

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

      1. *-un-lft-identity 60.7%

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

      2. add-sqr-sqrt 60.6%

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

      3. times-frac 60.6%

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

   4. Applied egg-rr 95.4%

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

      1. associate-*l/ 95.5%

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

      2. *-un-lft-identity 95.5%

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

      3. associate-/r* 95.5%

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

      4. associate-/l/ 94.5%

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

      5. *-commutative 94.5%

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

   6. Applied egg-rr 94.5%

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

   if 7.9999999999999993e-198 < c

   1. Initial program 72.1%

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

      1. add-cube-cbrt 72.0%

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

      2. pow3 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-*r* 64.5%

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

      5. unpow2 64.5%

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

      6. *-commutative 64.5%

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

      7. pow-prod-down 81.9%

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

      8. pow-prod-down 97.6%

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

      9. *-commutative 97.6%

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

   4. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}}\right)}^{3}} \]
   5. Step-by-step derivation

      1. /-rgt-identity 97.6%

         \[\leadsto {\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\frac{c \cdot \left(x \cdot s\right)}{1}\right)}}^{2}}}\right)}^{3} \]

      2. associate-*r* 95.4%

         \[\leadsto {\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\left(\frac{\color{blue}{\left(c \cdot x\right) \cdot s}}{1}\right)}^{2}}}\right)}^{3} \]

      3. associate-/l* 95.4%

         \[\leadsto {\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\frac{c \cdot x}{\frac{1}{s}}\right)}}^{2}}}\right)}^{3} \]

   6. Applied egg-rr 95.4%

      \[\leadsto {\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\frac{c \cdot x}{\frac{1}{s}}\right)}}^{2}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-/r/ 95.9%

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

      4. unpow-prod-down 80.8%

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

      5. /-rgt-identity 80.8%

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

      6. unpow-prod-down 95.9%

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

      7. associate-*r* 98.1%

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

      8. pow2 98.1%

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

      9. *-commutative 98.1%

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

      10. associate-*r* 95.8%

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

      11. associate-/l/ 96.7%

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

      12. associate-/r* 97.9%

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

      13. *-commutative 97.9%

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

      14. associate-*r* 94.7%

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

      15. *-commutative 94.7%

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

      16. associate-*l* 96.1%

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

   8. Applied egg-rr 96.1%

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

      1. expm1-log1p-u 80.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{x \cdot \left(c \cdot s\right)}\right)\right)}}{x \cdot s} \]

      2. expm1-udef 61.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{x \cdot \left(c \cdot s\right)}\right)} - 1}}{x \cdot s} \]

   10. Applied egg-rr 62.5%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot c}\right)} - 1}}{x \cdot s} \]
   11. Step-by-step derivation

       1. expm1-def 78.9%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot c}\right)\right)}}{x \cdot s} \]

       2. expm1-log1p 94.4%

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

       3. *-commutative 94.4%

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

       4. associate-*r* 97.7%

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

       5. *-commutative 97.7%

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

   12. Simplified 97.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{\left(s \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(c \cdot \left(s \cdot x\right)\right)}}{s \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{\cos \left(2 \cdot x\right)}{c_m}}{s_m \cdot \left(x \cdot \left(c_m \cdot \left(s_m \cdot x\right)\right)\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ (cos (* 2.0 x)) c_m) (* s_m (* x (* c_m (* s_m x))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (cos((2.0 * x)) / c_m) / (s_m * (x * (c_m * (s_m * x))));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (cos((2.0d0 * x)) / c_m) / (s_m * (x * (c_m * (s_m * x))))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (Math.cos((2.0 * x)) / c_m) / (s_m * (x * (c_m * (s_m * x))));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (math.cos((2.0 * x)) / c_m) / (s_m * (x * (c_m * (s_m * x))))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(cos(Float64(2.0 * x)) / c_m) / Float64(s_m * Float64(x * Float64(c_m * Float64(s_m * x)))))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (cos((2.0 * x)) / c_m) / (s_m * (x * (c_m * (s_m * x))));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(s$95$m * N[(x * N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. add-sqr-sqrt 65.1%

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

   3. times-frac 65.1%

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

4. Applied egg-rr 96.8%

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

   1. associate-*l/ 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. associate-/r* 96.9%

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

   4. associate-/l/ 95.4%

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

   5. *-commutative 95.4%

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

6. Applied egg-rr 95.4%

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

   1. *-commutative 95.4%

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

   2. *-commutative 95.4%

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

   3. associate-*r* 92.6%

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

   4. add-exp-log 45.8%

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

   5. add-exp-log 23.0%

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

   6. prod-exp 22.9%

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

   7. associate-*r* 23.0%

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

8. Applied egg-rr 23.0%

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

   1. exp-sum 23.0%

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

   2. rem-exp-log 47.5%

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

   3. rem-exp-log 91.2%

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

   4. associate-*l* 92.6%

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

   5. *-commutative 92.6%

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

10. Simplified 92.6%

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

11. Final simplification 92.6%

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

Alternative 6: 93.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{\cos \left(2 \cdot x\right)}{c_m}}{\left(s_m \cdot x\right) \cdot \left(c_m \cdot \left(s_m \cdot x\right)\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ (cos (* 2.0 x)) c_m) (* (* s_m x) (* c_m (* s_m x)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (cos((2.0 * x)) / c_m) / ((s_m * x) * (c_m * (s_m * x)));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (cos((2.0d0 * x)) / c_m) / ((s_m * x) * (c_m * (s_m * x)))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (Math.cos((2.0 * x)) / c_m) / ((s_m * x) * (c_m * (s_m * x)));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (math.cos((2.0 * x)) / c_m) / ((s_m * x) * (c_m * (s_m * x)))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(cos(Float64(2.0 * x)) / c_m) / Float64(Float64(s_m * x) * Float64(c_m * Float64(s_m * x))))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (cos((2.0 * x)) / c_m) / ((s_m * x) * (c_m * (s_m * x)));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(s$95$m * x), $MachinePrecision] * N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. add-sqr-sqrt 65.1%

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

   3. times-frac 65.1%

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

4. Applied egg-rr 96.8%

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

   1. associate-*l/ 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. associate-/r* 96.9%

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

   4. associate-/l/ 95.4%

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

   5. *-commutative 95.4%

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

6. Applied egg-rr 95.4%

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

7. Final simplification 95.4%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(s_m \cdot x\right)\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x)))) (/ (/ (cos (* 2.0 x)) t_0) t_0)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return (cos((2.0 * x)) / t_0) / t_0;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return (Math.cos((2.0 * x)) / t_0) / t_0;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	return (math.cos((2.0 * x)) / t_0) / t_0

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	return Float64(Float64(cos(Float64(2.0 * x)) / t_0) / t_0)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = (cos((2.0 * x)) / t_0) / t_0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. add-sqr-sqrt 65.1%

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

   3. times-frac 65.1%

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

4. Applied egg-rr 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l/ 96.8%

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

   3. div-inv 96.9%

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

   4. *-commutative 96.9%

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

6. Applied egg-rr 96.9%

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

7. Final simplification 96.9%

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

Alternative 8: 80.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ {\left(\frac{c_m}{\frac{\frac{1}{x}}{s_m}}\right)}^{-2} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m) :precision binary64 (pow (/ c_m (/ (/ 1.0 x) s_m)) -2.0))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return pow((c_m / ((1.0 / x) / s_m)), -2.0);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (c_m / ((1.0d0 / x) / s_m)) ** (-2.0d0)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return Math.pow((c_m / ((1.0 / x) / s_m)), -2.0);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return math.pow((c_m / ((1.0 / x) / s_m)), -2.0)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(c_m / Float64(Float64(1.0 / x) / s_m)) ^ -2.0
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (c_m / ((1.0 / x) / s_m)) ^ -2.0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[Power[N[(c$95$m / N[(N[(1.0 / x), $MachinePrecision] / s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 65.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.3%

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

   2. *-commutative 51.3%

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

   3. unpow2 51.3%

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

   4. unpow2 51.3%

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

   5. swap-sqr 63.6%

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

   6. unpow2 63.6%

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

   7. associate-/r* 63.8%

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

   8. unpow2 63.8%

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

   9. unpow2 63.8%

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

   10. swap-sqr 76.2%

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

   11. unpow2 76.2%

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

   12. *-commutative 76.2%

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

5. Simplified 76.2%

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

   1. pow-flip 76.4%

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

   2. *-commutative 76.4%

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

   3. metadata-eval 76.4%

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

7. Applied egg-rr 76.4%

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

   1. /-rgt-identity 76.4%

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

   2. associate-/l* 76.3%

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

9. Applied egg-rr 76.3%

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

10. Taylor expanded in x around 0 76.3%

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

    1. *-commutative 76.3%

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

    2. associate-/r* 76.2%

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

12. Simplified 76.2%

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

13. Final simplification 76.2%

    \[\leadsto {\left(\frac{c}{\frac{\frac{1}{x}}{s}}\right)}^{-2} \]
14. Add Preprocessing

Alternative 9: 80.0% accurate, 20.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{c_m \cdot \left(s_m \cdot x\right)} \cdot \frac{\frac{1}{s_m \cdot x}}{c_m} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (/ 1.0 (* c_m (* s_m x))) (/ (/ 1.0 (* s_m x)) c_m)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (s_m * x))) * ((1.0 / (s_m * x)) / c_m);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (s_m * x))) * ((1.0d0 / (s_m * x)) / c_m)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (s_m * x))) * ((1.0 / (s_m * x)) / c_m);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / (c_m * (s_m * x))) * ((1.0 / (s_m * x)) / c_m)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(1.0 / Float64(c_m * Float64(s_m * x))) * Float64(Float64(1.0 / Float64(s_m * x)) / c_m))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / (c_m * (s_m * x))) * ((1.0 / (s_m * x)) / c_m);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. add-sqr-sqrt 65.1%

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

   3. times-frac 65.1%

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

4. Applied egg-rr 96.8%

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

5. Taylor expanded in x around 0 76.4%

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

   1. associate-/r* 76.4%

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

   2. *-commutative 76.4%

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

   3. *-rgt-identity 76.4%

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

   4. associate-*r/ 76.3%

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

   5. associate-*l/ 76.3%

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

   6. *-lft-identity 76.3%

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

   7. *-commutative 76.3%

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

7. Simplified 76.3%

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

8. Final simplification 76.3%

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

Alternative 10: 76.6% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{\left(s_m \cdot c_m\right) \cdot \left(x \cdot \left(c_m \cdot \left(s_m \cdot x\right)\right)\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* s_m c_m) (* x (* c_m (* s_m x))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / ((s_m * c_m) * (x * (c_m * (s_m * x))));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((s_m * c_m) * (x * (c_m * (s_m * x))))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / ((s_m * c_m) * (x * (c_m * (s_m * x))));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / ((s_m * c_m) * (x * (c_m * (s_m * x))))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(Float64(s_m * c_m) * Float64(x * Float64(c_m * Float64(s_m * x)))))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / ((s_m * c_m) * (x * (c_m * (s_m * x))));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x * N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.3%

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

   2. *-commutative 51.3%

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

   3. unpow2 51.3%

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

   4. unpow2 51.3%

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

   5. swap-sqr 63.6%

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

   6. unpow2 63.6%

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

   7. associate-/r* 63.8%

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

   8. unpow2 63.8%

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

   9. unpow2 63.8%

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

   10. swap-sqr 76.2%

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

   11. unpow2 76.2%

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

   12. *-commutative 76.2%

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

5. Simplified 76.2%

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

   1. unpow2 76.2%

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

   2. associate-*r* 75.7%

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

   3. *-commutative 75.7%

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

   4. associate-*l* 74.8%

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

7. Applied egg-rr 74.8%

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

8. Final simplification 74.8%

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

Alternative 11: 76.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{\left(s_m \cdot c_m\right) \cdot \left(x \cdot \left(x \cdot \left(s_m \cdot c_m\right)\right)\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* s_m c_m) (* x (* x (* s_m c_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / ((s_m * c_m) * (x * (x * (s_m * c_m))));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((s_m * c_m) * (x * (x * (s_m * c_m))))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / ((s_m * c_m) * (x * (x * (s_m * c_m))));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / ((s_m * c_m) * (x * (x * (s_m * c_m))))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(Float64(s_m * c_m) * Float64(x * Float64(x * Float64(s_m * c_m)))))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / ((s_m * c_m) * (x * (x * (s_m * c_m))));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x * N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.3%

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

   2. *-commutative 51.3%

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

   3. unpow2 51.3%

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

   4. unpow2 51.3%

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

   5. swap-sqr 63.6%

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

   6. unpow2 63.6%

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

   7. associate-/r* 63.8%

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

   8. unpow2 63.8%

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

   9. unpow2 63.8%

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

   10. swap-sqr 76.2%

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

   11. unpow2 76.2%

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

   12. *-commutative 76.2%

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

5. Simplified 76.2%

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

   1. unpow2 76.2%

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

   2. associate-*r* 75.7%

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

   3. *-commutative 75.7%

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

   4. associate-*l* 74.8%

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

7. Applied egg-rr 74.8%

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

   1. /-rgt-identity 74.8%

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

   2. *-commutative 74.8%

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

   3. associate-*r* 75.5%

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

   4. associate-/l* 75.5%

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

9. Applied egg-rr 75.5%

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

    1. div-inv 75.5%

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

    2. inv-pow 75.5%

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

    3. pow-flip 75.5%

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

    4. metadata-eval 75.5%

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

    5. pow1 75.5%

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

11. Applied egg-rr 75.5%

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

12. Final simplification 75.5%

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

Alternative 12: 79.8% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(s_m \cdot x\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x)))) (/ 1.0 (* t_0 t_0))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return 1.0 / (t_0 * t_0);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return 1.0 / (t_0 * t_0);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	return 1.0 / (t_0 * t_0)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.3%

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

   2. *-commutative 51.3%

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

   3. unpow2 51.3%

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

   4. unpow2 51.3%

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

   5. swap-sqr 63.6%

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

   6. unpow2 63.6%

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

   7. associate-/r* 63.8%

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

   8. unpow2 63.8%

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

   9. unpow2 63.8%

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

   10. swap-sqr 76.2%

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

   11. unpow2 76.2%

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

   12. *-commutative 76.2%

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

5. Simplified 76.2%

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

   1. *-commutative 76.2%

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

   2. *-commutative 76.2%

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

   3. *-commutative 76.2%

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

   4. unpow2 76.2%

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

7. Applied egg-rr 76.2%

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

8. Final simplification 76.2%

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

Alternative 13: 79.3% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{1}{c_m}}{\left(s_m \cdot x\right) \cdot \left(c_m \cdot \left(s_m \cdot x\right)\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ 1.0 c_m) (* (* s_m x) (* c_m (* s_m x)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / c_m) / ((s_m * x) * (c_m * (s_m * x)));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / c_m) / ((s_m * x) * (c_m * (s_m * x)))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / c_m) / ((s_m * x) * (c_m * (s_m * x)));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / c_m) / ((s_m * x) * (c_m * (s_m * x)))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(1.0 / c_m) / Float64(Float64(s_m * x) * Float64(c_m * Float64(s_m * x))))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / c_m) / ((s_m * x) * (c_m * (s_m * x)));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(s$95$m * x), $MachinePrecision] * N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. add-sqr-sqrt 65.1%

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

   3. times-frac 65.1%

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

4. Applied egg-rr 96.8%

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

   1. associate-*l/ 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. associate-/r* 96.9%

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

   4. associate-/l/ 95.4%

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

   5. *-commutative 95.4%

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

6. Applied egg-rr 95.4%

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

7. Taylor expanded in x around 0 76.0%

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

8. Final simplification 76.0%

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))