mixedcos

Percentage Accurate: 66.9% → 98.8%

Time: 14.6s

Alternatives: 12

Speedup: 24.1×

• 32.0% 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: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{c_m} \cdot \frac{\frac{\frac{1}{c_m}}{x_m \cdot s_m}}{x_m \cdot s_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{\frac{1}{{\left(s_m \cdot \left(x_m \cdot c_m\right)\right)}^{-2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.3e-110)
   (* (/ 1.0 c_m) (/ (/ (/ 1.0 c_m) (* x_m s_m)) (* x_m s_m)))
   (/ (cos (* x_m -2.0)) (/ 1.0 (pow (* s_m (* x_m c_m)) -2.0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.3e-110) {
		tmp = (1.0 / c_m) * (((1.0 / c_m) / (x_m * s_m)) / (x_m * s_m));
	} else {
		tmp = cos((x_m * -2.0)) / (1.0 / pow((s_m * (x_m * c_m)), -2.0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.3d-110) then
        tmp = (1.0d0 / c_m) * (((1.0d0 / c_m) / (x_m * s_m)) / (x_m * s_m))
    else
        tmp = cos((x_m * (-2.0d0))) / (1.0d0 / ((s_m * (x_m * c_m)) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.3e-110) {
		tmp = (1.0 / c_m) * (((1.0 / c_m) / (x_m * s_m)) / (x_m * s_m));
	} else {
		tmp = Math.cos((x_m * -2.0)) / (1.0 / Math.pow((s_m * (x_m * c_m)), -2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.3e-110:
		tmp = (1.0 / c_m) * (((1.0 / c_m) / (x_m * s_m)) / (x_m * s_m))
	else:
		tmp = math.cos((x_m * -2.0)) / (1.0 / math.pow((s_m * (x_m * c_m)), -2.0))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.3e-110)
		tmp = Float64(Float64(1.0 / c_m) * Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) / Float64(x_m * s_m)));
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(1.0 / (Float64(s_m * Float64(x_m * c_m)) ^ -2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.3e-110)
		tmp = (1.0 / c_m) * (((1.0 / c_m) / (x_m * s_m)) / (x_m * s_m));
	else
		tmp = cos((x_m * -2.0)) / (1.0 / ((s_m * (x_m * c_m)) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.3e-110], N[(N[(1.0 / c$95$m), $MachinePrecision] * N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[Power[N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999995e-110

   1. Initial program 64.9%

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

      1. associate-/r* 64.8%

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

      2. unpow2 64.8%

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

      3. sqr-neg 64.8%

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

      4. unpow2 64.8%

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

      5. associate-/r* 64.9%

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

      6. cos-neg 64.9%

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

      7. *-commutative 64.9%

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

      8. distribute-rgt-neg-in 64.9%

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

      9. metadata-eval 64.9%

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

      10. associate-*r* 67.8%

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

      11. *-commutative 67.8%

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

      12. unpow2 67.8%

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

      13. sqr-neg 67.8%

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

      14. associate-*l* 75.6%

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

      15. associate-*r* 76.2%

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

      16. associate-*r* 73.2%

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

      17. associate-*r* 65.2%

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

      18. unpow2 65.2%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in x around 0 51.4%

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

      1. associate-/r* 51.2%

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

      2. *-commutative 51.2%

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

      3. unpow2 51.2%

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

      4. unpow2 51.2%

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

      5. swap-sqr 67.9%

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

      6. unpow2 67.9%

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

      7. associate-/r* 68.0%

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

      8. unpow2 68.0%

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

      9. unpow2 68.0%

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

      10. swap-sqr 82.0%

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

      11. unpow2 82.0%

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

      12. *-commutative 82.0%

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

   6. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. pow2 82.0%

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

      3. associate-*r* 81.0%

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

      4. associate-*l* 76.5%

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

      5. *-commutative 76.5%

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

      6. associate-*r* 77.2%

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

      7. *-commutative 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. metadata-eval 77.2%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 79.9%

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

      5. frac-times 79.9%

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

      6. associate-*r/ 79.9%

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

      7. *-commutative 79.9%

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

      8. associate-*r* 78.2%

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

      9. times-frac 75.3%

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

      10. associate-*r* 78.2%

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

      11. *-commutative 78.2%

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

      12. associate-*r* 79.3%

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

      13. associate-/r* 79.3%

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

   10. Applied egg-rr 79.3%

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

   if 1.29999999999999995e-110 < x

   1. Initial program 72.9%

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

      1. associate-/r* 71.8%

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

      2. unpow2 71.8%

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

      3. sqr-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/r* 72.9%

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

      6. cos-neg 72.9%

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

      7. *-commutative 72.9%

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

      8. distribute-rgt-neg-in 72.9%

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

      9. metadata-eval 72.9%

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

      10. associate-*r* 73.1%

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

      11. *-commutative 73.1%

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

      12. unpow2 73.1%

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

      13. sqr-neg 73.1%

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

      14. associate-*l* 75.2%

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

      15. associate-*r* 80.7%

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

      16. associate-*r* 78.6%

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

      17. associate-*r* 77.4%

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

      18. unpow2 77.4%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 71.8%

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

      1. associate-/r* 70.7%

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

      2. *-commutative 70.7%

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

      3. unpow2 70.7%

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

      4. unpow2 70.7%

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

      5. swap-sqr 76.4%

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

      6. unpow2 76.4%

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

      7. associate-/r* 77.5%

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

      8. *-commutative 77.5%

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

      9. unpow2 77.5%

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

      10. unpow2 77.5%

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

      11. swap-sqr 96.5%

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

      12. unpow2 96.5%

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

      13. *-commutative 96.5%

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

   6. Simplified 96.5%

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

      1. /-rgt-identity 96.5%

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

      2. clear-num 96.5%

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

      3. pow-flip 96.5%

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

      4. associate-*r* 94.4%

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

      5. *-commutative 94.4%

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

      6. metadata-eval 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{\frac{1}{c}}{x \cdot s}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}}}\\ \end{array} \]

Alternative 2: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{s_m}}{c_m}\\ \mathbf{if}\;x_m \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{{\left(c_m \cdot \left(x_m \cdot s_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_0}{x_m}\right) \cdot \frac{\cos \left(x_m \cdot 2\right)}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 s_m) c_m)))
   (if (<= x_m 4e+273)
     (/ (cos (* x_m -2.0)) (pow (* c_m (* x_m s_m)) 2.0))
     (* (* t_0 (/ t_0 x_m)) (/ (cos (* x_m 2.0)) x_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (1.0 / s_m) / c_m;
	double tmp;
	if (x_m <= 4e+273) {
		tmp = cos((x_m * -2.0)) / pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = (t_0 * (t_0 / x_m)) * (cos((x_m * 2.0)) / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / s_m) / c_m
    if (x_m <= 4d+273) then
        tmp = cos((x_m * (-2.0d0))) / ((c_m * (x_m * s_m)) ** 2.0d0)
    else
        tmp = (t_0 * (t_0 / x_m)) * (cos((x_m * 2.0d0)) / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (1.0 / s_m) / c_m;
	double tmp;
	if (x_m <= 4e+273) {
		tmp = Math.cos((x_m * -2.0)) / Math.pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = (t_0 * (t_0 / x_m)) * (Math.cos((x_m * 2.0)) / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (1.0 / s_m) / c_m
	tmp = 0
	if x_m <= 4e+273:
		tmp = math.cos((x_m * -2.0)) / math.pow((c_m * (x_m * s_m)), 2.0)
	else:
		tmp = (t_0 * (t_0 / x_m)) * (math.cos((x_m * 2.0)) / x_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(1.0 / s_m) / c_m)
	tmp = 0.0
	if (x_m <= 4e+273)
		tmp = Float64(cos(Float64(x_m * -2.0)) / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0));
	else
		tmp = Float64(Float64(t_0 * Float64(t_0 / x_m)) * Float64(cos(Float64(x_m * 2.0)) / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (1.0 / s_m) / c_m;
	tmp = 0.0;
	if (x_m <= 4e+273)
		tmp = cos((x_m * -2.0)) / ((c_m * (x_m * s_m)) ^ 2.0);
	else
		tmp = (t_0 * (t_0 / x_m)) * (cos((x_m * 2.0)) / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 4e+273], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$0 / x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999978e273

   1. Initial program 68.0%

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

      1. associate-/r* 67.5%

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

      2. unpow2 67.5%

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

      3. sqr-neg 67.5%

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

      4. unpow2 67.5%

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

      5. associate-/r* 68.0%

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

      6. cos-neg 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. associate-*r* 70.0%

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

      11. *-commutative 70.0%

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

      12. unpow2 70.0%

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

      13. sqr-neg 70.0%

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

      14. associate-*l* 75.6%

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

      15. associate-*r* 78.0%

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

      16. associate-*r* 75.2%

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

      17. associate-*r* 69.4%

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

      18. unpow2 69.4%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in x around inf 61.9%

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

      1. associate-/r* 61.4%

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

      2. *-commutative 61.4%

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

      3. unpow2 61.4%

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

      4. unpow2 61.4%

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

      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* 78.6%

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

      8. *-commutative 78.6%

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

      9. unpow2 78.6%

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

      10. unpow2 78.6%

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

      11. swap-sqr 96.6%

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

      12. unpow2 96.6%

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

      13. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if 3.99999999999999978e273 < x

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. unpow2 57.1%

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

      5. associate-/r* 57.1%

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

      6. cos-neg 57.1%

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

      7. *-commutative 57.1%

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

      8. distribute-rgt-neg-in 57.1%

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

      9. metadata-eval 57.1%

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

      10. associate-*r* 57.1%

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

      11. *-commutative 57.1%

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

      12. unpow2 57.1%

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

      13. sqr-neg 57.1%

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

      14. associate-*l* 71.4%

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

      15. associate-*r* 71.4%

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

      16. associate-*r* 71.4%

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

      17. associate-*r* 71.4%

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

      18. unpow2 71.4%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in x around inf 57.1%

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

      1. associate-/r* 57.1%

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

      2. *-commutative 57.1%

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

      3. unpow2 57.1%

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

      4. unpow2 57.1%

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

      5. swap-sqr 71.4%

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

      6. unpow2 71.4%

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

      7. associate-/r* 71.4%

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

      8. *-commutative 71.4%

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

      9. unpow2 71.4%

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

      10. unpow2 71.4%

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

      11. swap-sqr 72.1%

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

      12. unpow2 72.1%

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

      13. *-commutative 72.1%

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

   6. Simplified 72.1%

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

      1. add-sqr-sqrt 42.9%

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

      2. add-sqr-sqrt 72.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. swap-sqr 0.0%

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

      6. metadata-eval 0.0%

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

      7. metadata-eval 0.0%

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

      8. swap-sqr 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

      11. sqrt-unprod 72.1%

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

      12. add-sqr-sqrt 72.1%

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

      13. associate-*r* 85.5%

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

      14. pow-prod-down 71.4%

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

      15. associate-/r* 71.4%

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

      16. *-un-lft-identity 71.4%

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

      17. associate-*l/ 71.4%

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

      18. unpow2 71.4%

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

   8. Applied egg-rr 71.8%

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

      1. metadata-eval 71.8%

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

      2. pow-prod-up 71.8%

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

      3. inv-pow 71.8%

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

      4. associate-/l/ 71.8%

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

      5. inv-pow 71.8%

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

      6. associate-/l/ 71.8%

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

      7. *-un-lft-identity 71.8%

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

      8. times-frac 85.5%

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

   10. Applied egg-rr 85.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{s}}{c} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\right) \cdot \frac{\cos \left(x \cdot 2\right)}{x}\\ \end{array} \]

Alternative 3: 96.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{s_m}}{c_m}\\ t_1 := c_m \cdot \left(x_m \cdot s_m\right)\\ t_2 := \cos \left(x_m \cdot 2\right)\\ \mathbf{if}\;x_m \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{1}{t_1} \cdot \frac{t_2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_0}{x_m}\right) \cdot \frac{t_2}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 s_m) c_m))
        (t_1 (* c_m (* x_m s_m)))
        (t_2 (cos (* x_m 2.0))))
   (if (<= x_m 4e+273)
     (* (/ 1.0 t_1) (/ t_2 t_1))
     (* (* t_0 (/ t_0 x_m)) (/ t_2 x_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (1.0 / s_m) / c_m;
	double t_1 = c_m * (x_m * s_m);
	double t_2 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4e+273) {
		tmp = (1.0 / t_1) * (t_2 / t_1);
	} else {
		tmp = (t_0 * (t_0 / x_m)) * (t_2 / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 / s_m) / c_m
    t_1 = c_m * (x_m * s_m)
    t_2 = cos((x_m * 2.0d0))
    if (x_m <= 4d+273) then
        tmp = (1.0d0 / t_1) * (t_2 / t_1)
    else
        tmp = (t_0 * (t_0 / x_m)) * (t_2 / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (1.0 / s_m) / c_m;
	double t_1 = c_m * (x_m * s_m);
	double t_2 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4e+273) {
		tmp = (1.0 / t_1) * (t_2 / t_1);
	} else {
		tmp = (t_0 * (t_0 / x_m)) * (t_2 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (1.0 / s_m) / c_m
	t_1 = c_m * (x_m * s_m)
	t_2 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 4e+273:
		tmp = (1.0 / t_1) * (t_2 / t_1)
	else:
		tmp = (t_0 * (t_0 / x_m)) * (t_2 / x_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(1.0 / s_m) / c_m)
	t_1 = Float64(c_m * Float64(x_m * s_m))
	t_2 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 4e+273)
		tmp = Float64(Float64(1.0 / t_1) * Float64(t_2 / t_1));
	else
		tmp = Float64(Float64(t_0 * Float64(t_0 / x_m)) * Float64(t_2 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (1.0 / s_m) / c_m;
	t_1 = c_m * (x_m * s_m);
	t_2 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 4e+273)
		tmp = (1.0 / t_1) * (t_2 / t_1);
	else
		tmp = (t_0 * (t_0 / x_m)) * (t_2 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 4e+273], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$0 / x$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / x$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999978e273

   1. Initial program 68.0%

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

      1. associate-/r* 67.5%

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

      2. unpow2 67.5%

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

      3. sqr-neg 67.5%

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

      4. unpow2 67.5%

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

      5. associate-/r* 68.0%

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

      6. cos-neg 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. associate-*r* 70.0%

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

      11. *-commutative 70.0%

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

      12. unpow2 70.0%

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

      13. sqr-neg 70.0%

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

      14. associate-*l* 75.6%

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

      15. associate-*r* 78.0%

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

      16. associate-*r* 75.2%

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

      17. associate-*r* 69.4%

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

      18. unpow2 69.4%

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

   3. Simplified 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. add-sqr-sqrt 61.9%

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

      3. times-frac 61.8%

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

      4. sqrt-prod 61.9%

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

      5. unpow2 61.9%

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

      6. sqrt-prod 30.9%

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

      7. add-sqr-sqrt 44.0%

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

      8. pow-prod-down 44.0%

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

      9. sqrt-pow1 42.3%

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

      10. metadata-eval 42.3%

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

      11. pow1 42.3%

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

      12. *-commutative 42.3%

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

   5. Applied egg-rr 96.6%

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

   if 3.99999999999999978e273 < x

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. unpow2 57.1%

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

      5. associate-/r* 57.1%

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

      6. cos-neg 57.1%

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

      7. *-commutative 57.1%

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

      8. distribute-rgt-neg-in 57.1%

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

      9. metadata-eval 57.1%

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

      10. associate-*r* 57.1%

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

      11. *-commutative 57.1%

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

      12. unpow2 57.1%

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

      13. sqr-neg 57.1%

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

      14. associate-*l* 71.4%

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

      15. associate-*r* 71.4%

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

      16. associate-*r* 71.4%

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

      17. associate-*r* 71.4%

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

      18. unpow2 71.4%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in x around inf 57.1%

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

      1. associate-/r* 57.1%

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

      2. *-commutative 57.1%

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

      3. unpow2 57.1%

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

      4. unpow2 57.1%

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

      5. swap-sqr 71.4%

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

      6. unpow2 71.4%

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

      7. associate-/r* 71.4%

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

      8. *-commutative 71.4%

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

      9. unpow2 71.4%

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

      10. unpow2 71.4%

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

      11. swap-sqr 72.1%

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

      12. unpow2 72.1%

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

      13. *-commutative 72.1%

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

   6. Simplified 72.1%

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

      1. add-sqr-sqrt 42.9%

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

      2. add-sqr-sqrt 72.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. swap-sqr 0.0%

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

      6. metadata-eval 0.0%

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

      7. metadata-eval 0.0%

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

      8. swap-sqr 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

      11. sqrt-unprod 72.1%

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

      12. add-sqr-sqrt 72.1%

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

      13. associate-*r* 85.5%

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

      14. pow-prod-down 71.4%

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

      15. associate-/r* 71.4%

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

      16. *-un-lft-identity 71.4%

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

      17. associate-*l/ 71.4%

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

      18. unpow2 71.4%

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

   8. Applied egg-rr 71.8%

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

      1. metadata-eval 71.8%

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

      2. pow-prod-up 71.8%

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

      3. inv-pow 71.8%

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

      4. associate-/l/ 71.8%

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

      5. inv-pow 71.8%

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

      6. associate-/l/ 71.8%

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

      7. *-un-lft-identity 71.8%

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

      8. times-frac 85.5%

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

   10. Applied egg-rr 85.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{s}}{c} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\right) \cdot \frac{\cos \left(x \cdot 2\right)}{x}\\ \end{array} \]

Alternative 4: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 3.8 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\cos \left(x_m \cdot 2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 3.8e+275)
     (* (/ 1.0 t_0) (/ (cos (* x_m 2.0)) t_0))
     (/ (cos (* x_m -2.0)) (* s_m (* (* x_m c_m) (* x_m (* c_m s_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 3.8e+275) {
		tmp = (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 3.8d+275) then
        tmp = (1.0d0 / t_0) * (cos((x_m * 2.0d0)) / t_0)
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 3.8e+275) {
		tmp = (1.0 / t_0) * (Math.cos((x_m * 2.0)) / t_0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 3.8e+275:
		tmp = (1.0 / t_0) * (math.cos((x_m * 2.0)) / t_0)
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 3.8e+275)
		tmp = Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x_m * 2.0)) / t_0));
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(x_m * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 3.8e+275)
		tmp = (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
	else
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.8e+275], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.80000000000000012e275

   1. Initial program 68.0%

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

      1. associate-/r* 67.5%

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

      2. unpow2 67.5%

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

      3. sqr-neg 67.5%

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

      4. unpow2 67.5%

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

      5. associate-/r* 68.0%

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

      6. cos-neg 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. associate-*r* 70.0%

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

      11. *-commutative 70.0%

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

      12. unpow2 70.0%

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

      13. sqr-neg 70.0%

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

      14. associate-*l* 75.6%

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

      15. associate-*r* 78.0%

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

      16. associate-*r* 75.2%

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

      17. associate-*r* 69.4%

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

      18. unpow2 69.4%

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

   3. Simplified 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. add-sqr-sqrt 61.9%

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

      3. times-frac 61.8%

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

      4. sqrt-prod 61.9%

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

      5. unpow2 61.9%

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

      6. sqrt-prod 30.9%

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

      7. add-sqr-sqrt 44.0%

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

      8. pow-prod-down 44.0%

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

      9. sqrt-pow1 42.3%

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

      10. metadata-eval 42.3%

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

      11. pow1 42.3%

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

      12. *-commutative 42.3%

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

   5. Applied egg-rr 96.6%

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

   if 3.80000000000000012e275 < x

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. unpow2 57.1%

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

      5. associate-/r* 57.1%

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

      6. cos-neg 57.1%

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

      7. *-commutative 57.1%

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

      8. distribute-rgt-neg-in 57.1%

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

      9. metadata-eval 57.1%

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

      10. associate-*r* 57.1%

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

      11. *-commutative 57.1%

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

      12. unpow2 57.1%

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

      13. sqr-neg 57.1%

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

      14. associate-*l* 71.4%

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

      15. associate-*r* 71.4%

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

      16. associate-*r* 71.4%

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

      17. associate-*r* 71.4%

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

      18. unpow2 71.4%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in x around inf 57.1%

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

      1. associate-/r* 57.1%

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

      2. *-commutative 57.1%

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

      3. unpow2 57.1%

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

      4. unpow2 57.1%

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

      5. swap-sqr 71.4%

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

      6. unpow2 71.4%

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

      7. associate-/r* 71.4%

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

      8. *-commutative 71.4%

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

      9. unpow2 71.4%

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

      10. unpow2 71.4%

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

      11. swap-sqr 72.1%

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

      12. unpow2 72.1%

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

      13. *-commutative 72.1%

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

   6. Simplified 72.1%

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

      1. *-commutative 72.1%

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

      2. pow2 72.1%

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

      3. associate-*r* 72.1%

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

      4. associate-*r* 72.1%

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

      5. *-commutative 72.1%

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

      6. associate-*r* 85.4%

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

      7. *-commutative 85.4%

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

   8. Applied egg-rr 85.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 5: 96.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 1.55e-29)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m -2.0)) (* s_m (* (* x_m c_m) t_0))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.55e-29) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * t_0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 1.55d-29) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * ((x_m * c_m) * t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.55e-29) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.55e-29:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * t_0))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.55e-29)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.55e-29)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.55e-29], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000013e-29

   1. Initial program 67.2%

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

      1. associate-/r* 67.1%

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

      2. unpow2 67.1%

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

      3. sqr-neg 67.1%

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

      4. unpow2 67.1%

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

      5. associate-/r* 67.2%

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

      6. cos-neg 67.2%

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

      7. *-commutative 67.2%

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

      8. distribute-rgt-neg-in 67.2%

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

      9. metadata-eval 67.2%

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

      10. associate-*r* 69.8%

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

      11. *-commutative 69.8%

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

      12. unpow2 69.8%

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

      13. sqr-neg 69.8%

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

      14. associate-*l* 76.7%

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

      15. associate-*r* 77.3%

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

      16. associate-*r* 74.6%

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

      17. associate-*r* 67.5%

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

      18. unpow2 67.5%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 55.1%

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

      2. *-commutative 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. swap-sqr 69.9%

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

      6. unpow2 69.9%

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

      7. associate-/r* 70.0%

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

      8. unpow2 70.0%

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

      9. unpow2 70.0%

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

      10. swap-sqr 84.0%

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

      11. unpow2 84.0%

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

      12. *-commutative 84.0%

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

   6. Simplified 84.0%

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

      1. expm1-log1p-u 82.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 72.0%

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

      3. pow-flip 72.0%

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

      4. associate-*r* 73.4%

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

      5. *-commutative 73.4%

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

      6. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. expm1-def 80.9%

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

      2. expm1-log1p 82.2%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

   10. Simplified 84.7%

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

       1. *-commutative 84.7%

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

       2. associate-*r* 82.2%

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

       3. metadata-eval 82.2%

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

       4. pow-flip 82.1%

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

       5. unpow2 82.1%

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

       6. associate-/r* 82.2%

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

       7. associate-*r* 81.2%

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

       8. *-commutative 81.2%

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

       9. associate-*l* 80.6%

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

       10. associate-*r* 83.2%

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

       11. *-commutative 83.2%

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

       12. associate-*l* 84.1%

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

   12. Applied egg-rr 84.1%

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

   if 1.55000000000000013e-29 < x

   1. Initial program 69.0%

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

      1. associate-/r* 67.6%

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

      2. unpow2 67.6%

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

      3. sqr-neg 67.6%

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

      4. unpow2 67.6%

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

      5. associate-/r* 69.0%

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

      6. cos-neg 69.0%

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

      7. *-commutative 69.0%

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

      8. distribute-rgt-neg-in 69.0%

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

      9. metadata-eval 69.0%

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

      10. associate-*r* 69.3%

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

      11. *-commutative 69.3%

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

      12. unpow2 69.3%

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

      13. sqr-neg 69.3%

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

      14. associate-*l* 72.1%

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

      15. associate-*r* 79.0%

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

      16. associate-*r* 76.4%

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

      17. associate-*r* 74.9%

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

      18. unpow2 74.9%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 67.6%

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

      1. associate-/r* 66.1%

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

      2. *-commutative 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

      5. swap-sqr 73.6%

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

      6. unpow2 73.6%

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

      7. associate-/r* 75.0%

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

      8. *-commutative 75.0%

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

      9. unpow2 75.0%

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

      10. unpow2 75.0%

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

      11. swap-sqr 95.4%

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

      12. unpow2 95.4%

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

      13. *-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. /-rgt-identity 95.4%

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

      2. clear-num 95.4%

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

      3. pow-flip 95.5%

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

      4. associate-*r* 92.7%

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

      5. *-commutative 92.7%

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

      6. metadata-eval 92.7%

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

   8. Applied egg-rr 92.7%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

   10. Simplified 98.1%

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

       1. pow-flip 98.1%

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

       2. *-commutative 98.1%

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

       3. associate-*r* 92.7%

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

       4. metadata-eval 92.7%

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

       5. unpow2 92.7%

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

       6. associate-*r* 92.8%

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

       7. *-commutative 92.8%

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

       8. associate-*r* 87.5%

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

       9. associate-*r* 92.9%

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

       10. *-commutative 92.9%

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

       11. associate-*l* 90.2%

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

   12. Applied egg-rr 90.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 6: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 8e-29)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m -2.0)) (* s_m (* (* x_m c_m) (* x_m (* c_m s_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 8e-29) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 8d-29) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 8e-29) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 8e-29:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 8e-29)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(x_m * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 8e-29)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-29], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999955e-29

   1. Initial program 67.2%

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

      1. associate-/r* 67.1%

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

      2. unpow2 67.1%

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

      3. sqr-neg 67.1%

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

      4. unpow2 67.1%

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

      5. associate-/r* 67.2%

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

      6. cos-neg 67.2%

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

      7. *-commutative 67.2%

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

      8. distribute-rgt-neg-in 67.2%

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

      9. metadata-eval 67.2%

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

      10. associate-*r* 69.8%

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

      11. *-commutative 69.8%

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

      12. unpow2 69.8%

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

      13. sqr-neg 69.8%

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

      14. associate-*l* 76.7%

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

      15. associate-*r* 77.3%

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

      16. associate-*r* 74.6%

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

      17. associate-*r* 67.5%

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

      18. unpow2 67.5%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 55.1%

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

      2. *-commutative 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. swap-sqr 69.9%

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

      6. unpow2 69.9%

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

      7. associate-/r* 70.0%

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

      8. unpow2 70.0%

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

      9. unpow2 70.0%

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

      10. swap-sqr 84.0%

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

      11. unpow2 84.0%

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

      12. *-commutative 84.0%

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

   6. Simplified 84.0%

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

      1. expm1-log1p-u 82.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 72.0%

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

      3. pow-flip 72.0%

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

      4. associate-*r* 73.4%

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

      5. *-commutative 73.4%

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

      6. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. expm1-def 80.9%

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

      2. expm1-log1p 82.2%

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

      3. associate-*r* 84.7%

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

      4. *-commutative 84.7%

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

   10. Simplified 84.7%

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

       1. *-commutative 84.7%

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

       2. associate-*r* 82.2%

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

       3. metadata-eval 82.2%

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

       4. pow-flip 82.1%

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

       5. unpow2 82.1%

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

       6. associate-/r* 82.2%

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

       7. associate-*r* 81.2%

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

       8. *-commutative 81.2%

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

       9. associate-*l* 80.6%

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

       10. associate-*r* 83.2%

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

       11. *-commutative 83.2%

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

       12. associate-*l* 84.1%

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

   12. Applied egg-rr 84.1%

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

   if 7.99999999999999955e-29 < x

   1. Initial program 69.0%

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

      1. associate-/r* 67.6%

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

      2. unpow2 67.6%

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

      3. sqr-neg 67.6%

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

      4. unpow2 67.6%

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

      5. associate-/r* 69.0%

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

      6. cos-neg 69.0%

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

      7. *-commutative 69.0%

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

      8. distribute-rgt-neg-in 69.0%

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

      9. metadata-eval 69.0%

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

      10. associate-*r* 69.3%

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

      11. *-commutative 69.3%

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

      12. unpow2 69.3%

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

      13. sqr-neg 69.3%

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

      14. associate-*l* 72.1%

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

      15. associate-*r* 79.0%

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

      16. associate-*r* 76.4%

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

      17. associate-*r* 74.9%

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

      18. unpow2 74.9%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 67.6%

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

      1. associate-/r* 66.1%

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

      2. *-commutative 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

      5. swap-sqr 73.6%

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

      6. unpow2 73.6%

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

      7. associate-/r* 75.0%

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

      8. *-commutative 75.0%

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

      9. unpow2 75.0%

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

      10. unpow2 75.0%

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

      11. swap-sqr 95.4%

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

      12. unpow2 95.4%

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

      13. *-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. *-commutative 95.4%

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

      2. pow2 95.4%

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

      3. associate-*r* 95.5%

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

      4. associate-*r* 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*r* 87.5%

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

      7. *-commutative 87.5%

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

   8. Applied egg-rr 87.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 7: 93.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. associate-/r* 67.2%

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

   2. unpow2 67.2%

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

   3. sqr-neg 67.2%

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

   4. unpow2 67.2%

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

   5. associate-/r* 67.7%

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

   6. cos-neg 67.7%

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

   7. *-commutative 67.7%

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

   8. distribute-rgt-neg-in 67.7%

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

   9. metadata-eval 67.7%

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

   10. associate-*r* 69.7%

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

   11. *-commutative 69.7%

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

   12. unpow2 69.7%

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

   13. sqr-neg 69.7%

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

   14. associate-*l* 75.5%

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

   15. associate-*r* 77.8%

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

   16. associate-*r* 75.1%

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

   17. associate-*r* 69.5%

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

   18. unpow2 69.5%

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

3. Simplified 61.7%

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

4. Taylor expanded in x around inf 61.7%

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

   1. associate-/r* 61.3%

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

   2. *-commutative 61.3%

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

   3. unpow2 61.3%

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

   4. unpow2 61.3%

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

   5. swap-sqr 77.9%

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

   6. unpow2 77.9%

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

   7. associate-/r* 78.4%

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

   8. *-commutative 78.4%

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

   9. unpow2 78.4%

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

   10. unpow2 78.4%

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

   11. swap-sqr 96.0%

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

   12. unpow2 96.0%

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

   13. *-commutative 96.0%

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

6. Simplified 96.0%

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

   1. /-rgt-identity 96.0%

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

   2. clear-num 96.0%

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

   3. pow-flip 96.0%

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

   4. associate-*r* 94.8%

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

   5. *-commutative 94.8%

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

   6. metadata-eval 94.8%

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

8. Applied egg-rr 94.8%

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

   1. associate-*r* 98.1%

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

   2. *-commutative 98.1%

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

10. Simplified 98.1%

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

    1. pow-flip 98.1%

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

    2. metadata-eval 98.1%

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

    3. *-commutative 98.1%

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

    4. associate-*r* 94.8%

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

    5. unpow2 94.8%

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

    6. *-commutative 94.8%

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

    7. associate-*r* 91.1%

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

    8. associate-*l* 90.3%

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

    9. associate-*r* 90.7%

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

    10. *-commutative 90.7%

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

    11. associate-*l* 88.5%

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

    12. associate-*r* 91.8%

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

    13. *-commutative 91.8%

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

    14. associate-*l* 92.8%

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

    15. *-commutative 92.8%

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

12. Applied egg-rr 92.8%

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

13. Final simplification 92.8%

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

Alternative 8: 77.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s_m \leq 1.3 \cdot 10^{+179}:\\ \;\;\;\;\frac{1}{\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(c_m \cdot \left(x_m \cdot s_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c_m \cdot s_m\right) \cdot \left(x_m \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= s_m 1.3e+179)
   (/ 1.0 (* (* x_m c_m) (* s_m (* c_m (* x_m s_m)))))
   (/ 1.0 (* (* c_m s_m) (* x_m (* x_m (* c_m s_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 1.3e+179) {
		tmp = 1.0 / ((x_m * c_m) * (s_m * (c_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (s_m <= 1.3d+179) then
        tmp = 1.0d0 / ((x_m * c_m) * (s_m * (c_m * (x_m * s_m))))
    else
        tmp = 1.0d0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 1.3e+179) {
		tmp = 1.0 / ((x_m * c_m) * (s_m * (c_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if s_m <= 1.3e+179:
		tmp = 1.0 / ((x_m * c_m) * (s_m * (c_m * (x_m * s_m))))
	else:
		tmp = 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (s_m <= 1.3e+179)
		tmp = Float64(1.0 / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(c_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(x_m * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (s_m <= 1.3e+179)
		tmp = 1.0 / ((x_m * c_m) * (s_m * (c_m * (x_m * s_m))));
	else
		tmp = 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[s$95$m, 1.3e+179], N[(1.0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.3000000000000001e179

   1. Initial program 68.0%

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

      1. associate-/r* 67.4%

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

      2. unpow2 67.4%

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

      3. sqr-neg 67.4%

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

      4. unpow2 67.4%

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

      5. associate-/r* 68.0%

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

      6. cos-neg 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. associate-*r* 70.1%

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

      11. *-commutative 70.1%

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

      12. unpow2 70.1%

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

      13. sqr-neg 70.1%

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

      14. associate-*l* 76.6%

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

      15. associate-*r* 78.3%

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

      16. associate-*r* 74.9%

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

      17. associate-*r* 69.8%

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

      18. unpow2 69.8%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in x around 0 56.3%

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

      1. associate-/r* 55.8%

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

      2. *-commutative 55.8%

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

      3. unpow2 55.8%

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

      4. unpow2 55.8%

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

      5. swap-sqr 65.7%

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

      6. unpow2 65.7%

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

      7. associate-/r* 66.2%

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

      8. unpow2 66.2%

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

      9. unpow2 66.2%

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

      10. swap-sqr 77.0%

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

      11. unpow2 77.0%

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

      12. *-commutative 77.0%

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

   6. Simplified 77.0%

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

      1. *-commutative 77.0%

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

      2. pow2 77.0%

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

      3. associate-*r* 76.3%

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

      4. associate-*l* 74.6%

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

      5. *-commutative 74.6%

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

      6. associate-*r* 74.9%

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

      7. *-commutative 74.9%

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

   8. Applied egg-rr 74.9%

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

   9. Taylor expanded in x around 0 74.6%

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

   if 1.3000000000000001e179 < s

   1. Initial program 65.4%

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

      1. associate-/r* 65.4%

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

      2. unpow2 65.4%

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

      3. sqr-neg 65.4%

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

      4. unpow2 65.4%

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

      5. associate-/r* 65.4%

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

      6. cos-neg 65.4%

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

      7. *-commutative 65.4%

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

      8. distribute-rgt-neg-in 65.4%

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

      9. metadata-eval 65.4%

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

      10. associate-*r* 65.4%

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

      11. *-commutative 65.4%

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

      12. unpow2 65.4%

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

      13. sqr-neg 65.4%

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

      14. associate-*l* 65.4%

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

      15. associate-*r* 73.0%

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

      16. associate-*r* 76.5%

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

      17. associate-*r* 66.4%

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

      18. unpow2 66.4%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in x around 0 56.5%

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

      1. associate-/r* 56.5%

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

      2. *-commutative 56.5%

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

      3. unpow2 56.5%

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

      4. unpow2 56.5%

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

      5. swap-sqr 80.5%

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

      6. unpow2 80.5%

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

      7. associate-/r* 80.6%

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

      8. unpow2 80.6%

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

      9. unpow2 80.6%

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

      10. swap-sqr 94.9%

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

      11. unpow2 94.9%

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

      12. *-commutative 94.9%

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

   6. Simplified 94.9%

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

      1. unpow2 94.9%

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

      2. associate-*r* 83.8%

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

      3. *-commutative 83.8%

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

      4. associate-*l* 83.9%

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

      5. *-commutative 83.9%

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

      6. associate-*r* 84.1%

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

      7. *-commutative 84.1%

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

   8. Applied egg-rr 84.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.3 \cdot 10^{+179}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 9: 78.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{c_m \cdot \left(\left(x_m \cdot s_m\right) \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 5.2e+70)
     (/ 1.0 (* c_m (* (* x_m s_m) (* x_m (* c_m s_m)))))
     (/ (/ -1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5.2e+70) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 5.2d+70) then
        tmp = 1.0d0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))))
    else
        tmp = ((-1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5.2e+70) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 5.2e+70:
		tmp = 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))))
	else:
		tmp = (-1.0 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 5.2e+70)
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * Float64(c_m * s_m)))));
	else
		tmp = Float64(Float64(-1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 5.2e+70)
		tmp = 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
	else
		tmp = (-1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.2e+70], N[(1.0 / N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000001e70

   1. Initial program 67.7%

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

      1. associate-/r* 67.6%

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

      2. unpow2 67.6%

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

      3. sqr-neg 67.6%

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

      4. unpow2 67.6%

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

      5. associate-/r* 67.7%

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

      6. cos-neg 67.7%

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

      7. *-commutative 67.7%

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

      8. distribute-rgt-neg-in 67.7%

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

      9. metadata-eval 67.7%

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

      10. associate-*r* 70.1%

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

      11. *-commutative 70.1%

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

      12. unpow2 70.1%

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

      13. sqr-neg 70.1%

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

      14. associate-*l* 76.2%

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

      15. associate-*r* 78.2%

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

      16. associate-*r* 75.7%

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

      17. associate-*r* 69.3%

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

      18. unpow2 69.3%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in x around 0 56.4%

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

      1. associate-/r* 56.3%

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

      2. *-commutative 56.3%

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

      3. unpow2 56.3%

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

      4. unpow2 56.3%

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

      5. swap-sqr 69.7%

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

      6. unpow2 69.7%

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

      7. associate-/r* 69.8%

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

      8. unpow2 69.8%

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

      9. unpow2 69.8%

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

      10. swap-sqr 83.2%

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

      11. unpow2 83.2%

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

      12. *-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. *-commutative 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-*r* 80.6%

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

      5. *-commutative 80.6%

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

      6. associate-*r* 77.4%

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

      7. *-commutative 77.4%

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

   8. Applied egg-rr 77.4%

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

   if 5.2000000000000001e70 < x

   1. Initial program 67.6%

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

      1. associate-/r* 65.6%

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

      2. unpow2 65.6%

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

      3. sqr-neg 65.6%

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

      4. unpow2 65.6%

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

      5. associate-/r* 67.6%

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

      6. cos-neg 67.6%

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

      7. *-commutative 67.6%

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

      8. distribute-rgt-neg-in 67.6%

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      9. metadata-eval 67.6%

         \[\leadsto \frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      10. associate-*r* 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {\left(-s\right)}^{2}\right)\right) \cdot x}} \]

      11. *-commutative 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left({\left(-s\right)}^{2} \cdot x\right)}\right) \cdot x} \]

      12. unpow2 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)} \cdot x\right)\right) \cdot x} \]

      13. sqr-neg 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)\right) \cdot x} \]

      14. associate-*l* 72.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x} \]

      15. associate-*r* 76.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot \left(s \cdot x\right)\right)} \cdot x} \]

      16. associate-*r* 72.2%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}} \]

      17. associate-*r* 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}} \]

      18. unpow2 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \left(s \cdot \color{blue}{{x}^{2}}\right)} \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 55.8%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{{x}^{2} \cdot \color{blue}{\left(s \cdot s\right)}} \]

      4. unpow2 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot s\right)} \]

      5. swap-sqr 56.0%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}} \]

      6. unpow2 56.0%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(x \cdot s\right)}^{2}}} \]

      7. associate-/r* 58.1%

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(x \cdot s\right)}^{2}}} \]

      8. unpow2 58.1%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}} \]

      9. unpow2 58.1%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      10. swap-sqr 59.6%

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      11. unpow2 59.6%

          \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

      12. *-commutative 59.6%

          \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

   6. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 59.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)} - 1} \]

      3. pow-flip 59.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. associate-*r* 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{\left(-2\right)}\right)} - 1 \]

      5. *-commutative 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{\left(-2\right)}\right)} - 1 \]

      6. metadata-eval 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   8. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 59.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 59.0%

         \[\leadsto \color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}} \]

      3. associate-*r* 59.8%

         \[\leadsto {\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}}^{-2} \]

      4. *-commutative 59.8%

         \[\leadsto {\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}^{-2} \]

   10. Simplified 59.8%

       \[\leadsto \color{blue}{{\left(\left(c \cdot x\right) \cdot s\right)}^{-2}} \]
   11. Step-by-step derivation

       1. *-commutative 59.8%

          \[\leadsto {\left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)}^{-2} \]

       2. associate-*r* 59.0%

          \[\leadsto {\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{-2} \]

       3. metadata-eval 59.0%

          \[\leadsto {\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{\left(-2\right)}} \]

       4. pow-flip 59.0%

          \[\leadsto \color{blue}{\frac{1}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

       5. frac-2neg 59.0%

          \[\leadsto \color{blue}{\frac{-1}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

       6. metadata-eval 59.0%

          \[\leadsto \frac{\color{blue}{-1}}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

       7. add-sqr-sqrt 4.3%

          \[\leadsto \frac{-1}{\color{blue}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       8. associate-/r* 4.3%

          \[\leadsto \color{blue}{\frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       9. add-sqr-sqrt 2.5%

          \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       10. sqrt-unprod 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{\left(-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}\right) \cdot \left(-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}\right)}}}} \]

       11. sqr-neg 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\sqrt{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2} \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       12. sqrt-unprod 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       13. add-sqr-sqrt 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       14. sqrt-pow1 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{\left(\frac{2}{2}\right)}}} \]

       15. metadata-eval 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{1}}} \]

       16. pow1 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   12. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{\frac{-1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 10: 79.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 5.2e+70) (/ (/ 1.0 t_0) t_0) (/ (/ -1.0 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5.2e+70) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 5.2d+70) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = ((-1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5.2e+70) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 5.2e+70:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (-1.0 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 5.2e+70)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(-1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 5.2e+70)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (-1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.2e+70], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000001e70

   1. Initial program 67.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 67.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. unpow2 67.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x} \]

      3. sqr-neg 67.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)}\right) \cdot x} \]

      4. unpow2 67.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{{\left(-s\right)}^{2}}\right) \cdot x} \]

      5. associate-/r* 67.7%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)}} \]

      6. cos-neg 67.7%

         \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      7. *-commutative 67.7%

         \[\leadsto \frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      8. distribute-rgt-neg-in 67.7%

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      9. metadata-eval 67.7%

         \[\leadsto \frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      10. associate-*r* 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {\left(-s\right)}^{2}\right)\right) \cdot x}} \]

      11. *-commutative 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left({\left(-s\right)}^{2} \cdot x\right)}\right) \cdot x} \]

      12. unpow2 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)} \cdot x\right)\right) \cdot x} \]

      13. sqr-neg 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)\right) \cdot x} \]

      14. associate-*l* 76.2%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x} \]

      15. associate-*r* 78.2%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot \left(s \cdot x\right)\right)} \cdot x} \]

      16. associate-*r* 75.7%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}} \]

      17. associate-*r* 69.3%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}} \]

      18. unpow2 69.3%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \left(s \cdot \color{blue}{{x}^{2}}\right)} \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 56.4%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 56.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 56.3%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 56.3%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{{x}^{2} \cdot \color{blue}{\left(s \cdot s\right)}} \]

      4. unpow2 56.3%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot s\right)} \]

      5. swap-sqr 69.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}} \]

      6. unpow2 69.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(x \cdot s\right)}^{2}}} \]

      7. associate-/r* 69.8%

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(x \cdot s\right)}^{2}}} \]

      8. unpow2 69.8%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}} \]

      9. unpow2 69.8%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      10. swap-sqr 83.2%

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      11. unpow2 83.2%

          \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

      12. *-commutative 83.2%

          \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

   6. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 82.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 72.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)} - 1} \]

      3. pow-flip 72.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. associate-*r* 73.2%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{\left(-2\right)}\right)} - 1 \]

      5. *-commutative 73.2%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{\left(-2\right)}\right)} - 1 \]

      6. metadata-eval 73.2%

         \[\leadsto e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   8. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 80.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 81.5%

         \[\leadsto \color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}} \]

      3. associate-*r* 83.8%

         \[\leadsto {\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}}^{-2} \]

      4. *-commutative 83.8%

         \[\leadsto {\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}^{-2} \]

   10. Simplified 83.8%

       \[\leadsto \color{blue}{{\left(\left(c \cdot x\right) \cdot s\right)}^{-2}} \]
   11. Step-by-step derivation

       1. *-commutative 83.8%

          \[\leadsto {\left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)}^{-2} \]

       2. associate-*r* 81.5%

          \[\leadsto {\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{-2} \]

       3. metadata-eval 81.5%

          \[\leadsto {\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{\left(-2\right)}} \]

       4. pow-flip 81.5%

          \[\leadsto \color{blue}{\frac{1}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

       5. unpow2 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

       6. associate-/r* 81.5%

          \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

       7. associate-*r* 80.7%

          \[\leadsto \frac{\frac{1}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

       8. *-commutative 80.7%

          \[\leadsto \frac{\frac{1}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

       9. associate-*l* 80.1%

          \[\leadsto \frac{\frac{1}{\color{blue}{c \cdot \left(x \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

       10. associate-*r* 82.4%

           \[\leadsto \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

       11. *-commutative 82.4%

           \[\leadsto \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

       12. associate-*l* 83.2%

           \[\leadsto \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{\color{blue}{c \cdot \left(x \cdot s\right)}} \]

   12. Applied egg-rr 83.2%

       \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}} \]

   if 5.2000000000000001e70 < x

   1. Initial program 67.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.6%

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

      2. unpow2 65.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x} \]

      3. sqr-neg 65.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)}\right) \cdot x} \]

      4. unpow2 65.6%

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{{\left(-s\right)}^{2}}\right) \cdot x} \]

      5. associate-/r* 67.6%

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)}} \]

      6. cos-neg 67.6%

         \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      7. *-commutative 67.6%

         \[\leadsto \frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      8. distribute-rgt-neg-in 67.6%

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      9. metadata-eval 67.6%

         \[\leadsto \frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

      10. associate-*r* 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {\left(-s\right)}^{2}\right)\right) \cdot x}} \]

      11. *-commutative 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left({\left(-s\right)}^{2} \cdot x\right)}\right) \cdot x} \]

      12. unpow2 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)} \cdot x\right)\right) \cdot x} \]

      13. sqr-neg 68.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)\right) \cdot x} \]

      14. associate-*l* 72.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x} \]

      15. associate-*r* 76.0%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot \left(s \cdot x\right)\right)} \cdot x} \]

      16. associate-*r* 72.2%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}} \]

      17. associate-*r* 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}} \]

      18. unpow2 70.1%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \left(s \cdot \color{blue}{{x}^{2}}\right)} \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 55.8%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{{x}^{2} \cdot \color{blue}{\left(s \cdot s\right)}} \]

      4. unpow2 53.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot s\right)} \]

      5. swap-sqr 56.0%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}} \]

      6. unpow2 56.0%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(x \cdot s\right)}^{2}}} \]

      7. associate-/r* 58.1%

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(x \cdot s\right)}^{2}}} \]

      8. unpow2 58.1%

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}} \]

      9. unpow2 58.1%

         \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      10. swap-sqr 59.6%

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      11. unpow2 59.6%

          \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

      12. *-commutative 59.6%

          \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

   6. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 59.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)} - 1} \]

      3. pow-flip 59.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. associate-*r* 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{\left(-2\right)}\right)} - 1 \]

      5. *-commutative 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{\left(-2\right)}\right)} - 1 \]

      6. metadata-eval 58.6%

         \[\leadsto e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   8. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 59.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 59.0%

         \[\leadsto \color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}} \]

      3. associate-*r* 59.8%

         \[\leadsto {\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}}^{-2} \]

      4. *-commutative 59.8%

         \[\leadsto {\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}^{-2} \]

   10. Simplified 59.8%

       \[\leadsto \color{blue}{{\left(\left(c \cdot x\right) \cdot s\right)}^{-2}} \]
   11. Step-by-step derivation

       1. *-commutative 59.8%

          \[\leadsto {\left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)}^{-2} \]

       2. associate-*r* 59.0%

          \[\leadsto {\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{-2} \]

       3. metadata-eval 59.0%

          \[\leadsto {\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{\left(-2\right)}} \]

       4. pow-flip 59.0%

          \[\leadsto \color{blue}{\frac{1}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

       5. frac-2neg 59.0%

          \[\leadsto \color{blue}{\frac{-1}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

       6. metadata-eval 59.0%

          \[\leadsto \frac{\color{blue}{-1}}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

       7. add-sqr-sqrt 4.3%

          \[\leadsto \frac{-1}{\color{blue}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       8. associate-/r* 4.3%

          \[\leadsto \color{blue}{\frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       9. add-sqr-sqrt 2.5%

          \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       10. sqrt-unprod 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{\left(-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}\right) \cdot \left(-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}\right)}}}} \]

       11. sqr-neg 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\sqrt{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2} \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       12. sqrt-unprod 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{\sqrt{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \cdot \sqrt{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}} \]

       13. add-sqr-sqrt 2.5%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\sqrt{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}} \]

       14. sqrt-pow1 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{\left(\frac{2}{2}\right)}}} \]

       15. metadata-eval 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{\color{blue}{1}}} \]

       16. pow1 0.4%

           \[\leadsto \frac{\frac{-1}{\sqrt{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   12. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{\frac{-1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 11: 76.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{\left(c_m \cdot s_m\right) \cdot \left(x_m \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* (* c_m s_m) (* x_m (* x_m (* c_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(x_m * Float64(c_m * s_m)))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 1.0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. unpow2 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x} \]

   3. sqr-neg 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)}\right) \cdot x} \]

   4. unpow2 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{{\left(-s\right)}^{2}}\right) \cdot x} \]

   5. associate-/r* 67.7%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)}} \]

   6. cos-neg 67.7%

      \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   7. *-commutative 67.7%

      \[\leadsto \frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   8. distribute-rgt-neg-in 67.7%

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   9. metadata-eval 67.7%

      \[\leadsto \frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   10. associate-*r* 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {\left(-s\right)}^{2}\right)\right) \cdot x}} \]

   11. *-commutative 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left({\left(-s\right)}^{2} \cdot x\right)}\right) \cdot x} \]

   12. unpow2 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)} \cdot x\right)\right) \cdot x} \]

   13. sqr-neg 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)\right) \cdot x} \]

   14. associate-*l* 75.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x} \]

   15. associate-*r* 77.8%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot \left(s \cdot x\right)\right)} \cdot x} \]

   16. associate-*r* 75.1%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}} \]

   17. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}} \]

   18. unpow2 69.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \left(s \cdot \color{blue}{{x}^{2}}\right)} \]

3. Simplified 61.7%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

4. Taylor expanded in x around 0 56.3%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
5. Step-by-step derivation

   1. associate-/r* 55.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{{x}^{2} \cdot \color{blue}{\left(s \cdot s\right)}} \]

   4. unpow2 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot s\right)} \]

   5. swap-sqr 67.1%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}} \]

   6. unpow2 67.1%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(x \cdot s\right)}^{2}}} \]

   7. associate-/r* 67.6%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(x \cdot s\right)}^{2}}} \]

   8. unpow2 67.6%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}} \]

   9. unpow2 67.6%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   10. swap-sqr 78.8%

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   11. unpow2 78.8%

       \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

   12. *-commutative 78.8%

       \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

6. Simplified 78.8%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 78.8%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

   2. associate-*r* 76.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot \left(s \cdot x\right)\right)} \]

   3. *-commutative 76.1%

      \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

   4. associate-*l* 73.7%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}} \]

   5. *-commutative 73.7%

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

   6. associate-*r* 74.0%

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

   7. *-commutative 74.0%

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right)} \]

8. Applied egg-rr 74.0%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}} \]

9. Final simplification 74.0%

   \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)} \]

Alternative 12: 77.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{\left(x_m \cdot s_m\right) \cdot \left(c_m \cdot \left(x_m \cdot \left(c_m \cdot s_m\right)\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* (* x_m s_m) (* c_m (* x_m (* c_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((x_m * s_m) * (c_m * (x_m * (c_m * s_m))));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((x_m * s_m) * (c_m * (x_m * (c_m * s_m))))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((x_m * s_m) * (c_m * (x_m * (c_m * s_m))));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / ((x_m * s_m) * (c_m * (x_m * (c_m * s_m))))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(1.0 / Float64(Float64(x_m * s_m) * Float64(c_m * Float64(x_m * Float64(c_m * s_m)))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 1.0 / ((x_m * s_m) * (c_m * (x_m * (c_m * s_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. unpow2 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x} \]

   3. sqr-neg 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)}\right) \cdot x} \]

   4. unpow2 67.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot \color{blue}{{\left(-s\right)}^{2}}\right) \cdot x} \]

   5. associate-/r* 67.7%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)}} \]

   6. cos-neg 67.7%

      \[\leadsto \frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   7. *-commutative 67.7%

      \[\leadsto \frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   8. distribute-rgt-neg-in 67.7%

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   9. metadata-eval 67.7%

      \[\leadsto \frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {\left(-s\right)}^{2}\right) \cdot x\right)} \]

   10. associate-*r* 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {\left(-s\right)}^{2}\right)\right) \cdot x}} \]

   11. *-commutative 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left({\left(-s\right)}^{2} \cdot x\right)}\right) \cdot x} \]

   12. unpow2 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(\left(-s\right) \cdot \left(-s\right)\right)} \cdot x\right)\right) \cdot x} \]

   13. sqr-neg 69.7%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)\right) \cdot x} \]

   14. associate-*l* 75.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \cdot x} \]

   15. associate-*r* 77.8%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot \left(s \cdot x\right)\right)} \cdot x} \]

   16. associate-*r* 75.1%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left({c}^{2} \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}} \]

   17. associate-*r* 69.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}} \]

   18. unpow2 69.5%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\left({c}^{2} \cdot s\right) \cdot \left(s \cdot \color{blue}{{x}^{2}}\right)} \]

3. Simplified 61.7%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

4. Taylor expanded in x around 0 56.3%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
5. Step-by-step derivation

   1. associate-/r* 55.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{{x}^{2} \cdot \color{blue}{\left(s \cdot s\right)}} \]

   4. unpow2 55.9%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot s\right)} \]

   5. swap-sqr 67.1%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}} \]

   6. unpow2 67.1%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(x \cdot s\right)}^{2}}} \]

   7. associate-/r* 67.6%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(x \cdot s\right)}^{2}}} \]

   8. unpow2 67.6%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}} \]

   9. unpow2 67.6%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   10. swap-sqr 78.8%

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   11. unpow2 78.8%

       \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

   12. *-commutative 78.8%

       \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{2}} \]

6. Simplified 78.8%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 78.8%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

   2. *-commutative 78.8%

      \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(x \cdot s\right)}\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)} \]

   3. associate-*r* 77.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot \left(x \cdot s\right)\right) \cdot c\right) \cdot \left(s \cdot x\right)}} \]

   4. *-commutative 77.1%

      \[\leadsto \frac{1}{\left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot c\right) \cdot \left(s \cdot x\right)} \]

   5. associate-*r* 75.2%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot c\right) \cdot \left(s \cdot x\right)} \]

   6. *-commutative 75.2%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot c\right) \cdot \left(s \cdot x\right)} \]

8. Applied egg-rr 75.2%

   \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot c\right) \cdot \left(s \cdot x\right)}} \]

9. Final simplification 75.2%

   \[\leadsto \frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)} \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))