mixedcos

Percentage Accurate: 65.9% → 98.1%

Time: 15.7s

Alternatives: 11

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% 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: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x_m \cdot -2\right)\\ \mathbf{if}\;x_m \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{t_0}{c_m \cdot \left(\left(x_m \cdot s_m\right) \cdot \left(c_m \cdot \left(x_m \cdot s_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x_m -2.0))))
   (if (<= x_m 4.4e-36)
     (/ t_0 (* c_m (* (* x_m s_m) (* c_m (* x_m s_m)))))
     (/ t_0 (* s_m (* (* x_m c_m) (* s_m (* x_m c_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 = cos((x_m * -2.0));
	double tmp;
	if (x_m <= 4.4e-36) {
		tmp = t_0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 = cos((x_m * (-2.0d0)))
    if (x_m <= 4.4d-36) then
        tmp = t_0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
    else
        tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 = Math.cos((x_m * -2.0));
	double tmp;
	if (x_m <= 4.4e-36) {
		tmp = t_0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 = math.cos((x_m * -2.0))
	tmp = 0
	if x_m <= 4.4e-36:
		tmp = t_0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
	else:
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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 = cos(Float64(x_m * -2.0))
	tmp = 0.0
	if (x_m <= 4.4e-36)
		tmp = Float64(t_0 / Float64(c_m * Float64(Float64(x_m * s_m) * Float64(c_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_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 = cos((x_m * -2.0));
	tmp = 0.0;
	if (x_m <= 4.4e-36)
		tmp = t_0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	else
		tmp = t_0 / (s_m * ((x_m * c_m) * (s_m * (x_m * c_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[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 4.4e-36], N[(t$95$0 / 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], N[(t$95$0 / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3999999999999999e-36

   1. Initial program 72.2%

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

      1. associate-/r* 71.6%

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

      2. associate-*l* 71.6%

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

      3. unpow2 71.6%

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

      4. sqr-neg 71.6%

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

      5. unpow2 71.6%

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

      6. *-commutative 71.6%

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

      7. *-commutative 71.6%

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

      8. associate-/r* 72.2%

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

      9. cos-neg 72.2%

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

      10. *-commutative 72.2%

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

      11. distribute-rgt-neg-in 72.2%

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

      12. metadata-eval 72.2%

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

      13. associate-*r* 72.8%

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

      14. *-commutative 72.8%

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

      15. unpow2 72.8%

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

      16. sqr-neg 72.8%

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

      17. associate-*l* 81.6%

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

      18. associate-*r* 81.5%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 66.3%

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

      1. associate-/r* 65.7%

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

      2. *-commutative 65.7%

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

      3. unpow2 65.7%

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

      4. unpow2 65.7%

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

      5. swap-sqr 83.1%

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

      6. unpow2 83.1%

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

      7. associate-/r* 84.1%

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

      8. *-commutative 84.1%

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

      9. unpow2 84.1%

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

      10. unpow2 84.1%

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

      11. swap-sqr 98.6%

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

      12. unpow2 98.6%

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

      13. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. unpow-prod-down 84.1%

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

      2. *-commutative 84.1%

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

      3. pow-prod-down 98.6%

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

      4. pow2 98.6%

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

      5. *-commutative 98.6%

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

      6. associate-*r* 97.2%

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

      7. *-commutative 97.2%

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

      8. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 4.3999999999999999e-36 < x

   1. Initial program 70.2%

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

      1. associate-/r* 69.3%

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

      2. associate-*l* 69.3%

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

      3. unpow2 69.3%

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

      4. sqr-neg 69.3%

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

      5. unpow2 69.3%

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

      6. *-commutative 69.3%

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

      7. *-commutative 69.3%

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

      8. associate-/r* 70.2%

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

      9. cos-neg 70.2%

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

      10. *-commutative 70.2%

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

      11. distribute-rgt-neg-in 70.2%

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

      12. metadata-eval 70.2%

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

      13. associate-*r* 71.9%

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

      14. *-commutative 71.9%

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

      15. unpow2 71.9%

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

      16. sqr-neg 71.9%

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

      17. associate-*l* 77.0%

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

      18. associate-*r* 81.0%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in x around inf 62.9%

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

      1. associate-/r* 62.8%

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

      2. *-commutative 62.8%

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

      3. unpow2 62.8%

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

      4. unpow2 62.8%

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

      5. swap-sqr 77.1%

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

      6. unpow2 77.1%

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

      7. associate-/r* 78.0%

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

      8. *-commutative 78.0%

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

      9. unpow2 78.0%

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

      10. unpow2 78.0%

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

      11. swap-sqr 99.6%

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

      12. unpow2 99.6%

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

      13. associate-*r* 97.2%

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

      14. *-commutative 97.2%

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

   7. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. *-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. associate-*r* 97.3%

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

      5. associate-*r* 94.8%

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

      6. associate-*r* 94.8%

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

      7. *-commutative 94.8%

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

   9. Applied egg-rr 94.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.5× speedup

Math

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

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)) (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) {
	return cos((x_m * -2.0)) / pow((s_m * (x_m * c_m)), 2.0);
}

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))) / ((s_m * (x_m * c_m)) ** 2.0d0)
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)) / Math.pow((s_m * (x_m * c_m)), 2.0);
}

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)) / math.pow((s_m * (x_m * c_m)), 2.0)

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(s_m * Float64(x_m * c_m)) ^ 2.0))
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)) / ((s_m * (x_m * c_m)) ^ 2.0);
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[Power[N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. associate-/r* 70.9%

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

   2. associate-*l* 70.9%

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

   3. unpow2 70.9%

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

   4. sqr-neg 70.9%

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

   5. unpow2 70.9%

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

   6. *-commutative 70.9%

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

   7. *-commutative 70.9%

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

   8. associate-/r* 71.6%

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

   9. cos-neg 71.6%

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

   10. *-commutative 71.6%

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

   11. distribute-rgt-neg-in 71.6%

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

   12. metadata-eval 71.6%

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

   13. associate-*r* 72.5%

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

   14. *-commutative 72.5%

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

   15. unpow2 72.5%

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

   16. sqr-neg 72.5%

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

   17. associate-*l* 80.2%

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

   18. associate-*r* 81.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 65.3%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. unpow2 64.9%

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

   4. unpow2 64.9%

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

   5. swap-sqr 81.3%

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

   6. unpow2 81.3%

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

   7. associate-/r* 82.2%

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

   8. *-commutative 82.2%

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

   9. unpow2 82.2%

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

   10. unpow2 82.2%

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

   11. swap-sqr 98.9%

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

   12. unpow2 98.9%

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

   13. associate-*r* 97.7%

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

   14. *-commutative 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 3: 96.6% accurate, 1.5× speedup

Math

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

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)) (pow (* c_m (* x_m s_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) {
	return cos((x_m * -2.0)) / pow((c_m * (x_m * s_m)), 2.0);
}

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)) ** 2.0d0)
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)) / Math.pow((c_m * (x_m * s_m)), 2.0);
}

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)) / math.pow((c_m * (x_m * s_m)), 2.0)

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(x_m * s_m)) ^ 2.0))
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)) ^ 2.0);
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[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. associate-/r* 70.9%

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

   2. associate-*l* 70.9%

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

   3. unpow2 70.9%

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

   4. sqr-neg 70.9%

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

   5. unpow2 70.9%

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

   6. *-commutative 70.9%

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

   7. *-commutative 70.9%

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

   8. associate-/r* 71.6%

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

   9. cos-neg 71.6%

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

   10. *-commutative 71.6%

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

   11. distribute-rgt-neg-in 71.6%

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

   12. metadata-eval 71.6%

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

   13. associate-*r* 72.5%

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

   14. *-commutative 72.5%

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

   15. unpow2 72.5%

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

   16. sqr-neg 72.5%

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

   17. associate-*l* 80.2%

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

   18. associate-*r* 81.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 65.3%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. unpow2 64.9%

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

   4. unpow2 64.9%

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

   5. swap-sqr 81.3%

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

   6. unpow2 81.3%

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

   7. associate-/r* 82.2%

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

   8. *-commutative 82.2%

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

   9. unpow2 82.2%

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

   10. unpow2 82.2%

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

   11. swap-sqr 98.9%

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

   12. unpow2 98.9%

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

   13. *-commutative 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 95.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 5.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{c_m}}{\left(x_m \cdot s_m\right) \cdot \left(c_m \cdot \left(x_m \cdot s_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{\left(x_m \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right) \cdot \left(s_m \cdot c_m\right)}\\ \end{array} \end{array} \]

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 (<= x_m 5.2e-100)
   (/ (/ 1.0 c_m) (* (* x_m s_m) (* c_m (* x_m s_m))))
   (/ (cos (* x_m -2.0)) (* (* x_m (* s_m (* x_m c_m))) (* s_m c_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 (x_m <= 5.2e-100) {
		tmp = (1.0 / c_m) / ((x_m * s_m) * (c_m * (x_m * s_m)));
	} else {
		tmp = cos((x_m * -2.0)) / ((x_m * (s_m * (x_m * c_m))) * (s_m * c_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 (x_m <= 5.2d-100) then
        tmp = (1.0d0 / c_m) / ((x_m * s_m) * (c_m * (x_m * s_m)))
    else
        tmp = cos((x_m * (-2.0d0))) / ((x_m * (s_m * (x_m * c_m))) * (s_m * c_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 (x_m <= 5.2e-100) {
		tmp = (1.0 / c_m) / ((x_m * s_m) * (c_m * (x_m * s_m)));
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((x_m * (s_m * (x_m * c_m))) * (s_m * c_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 x_m <= 5.2e-100:
		tmp = (1.0 / c_m) / ((x_m * s_m) * (c_m * (x_m * s_m)))
	else:
		tmp = math.cos((x_m * -2.0)) / ((x_m * (s_m * (x_m * c_m))) * (s_m * c_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 (x_m <= 5.2e-100)
		tmp = Float64(Float64(1.0 / c_m) / Float64(Float64(x_m * s_m) * Float64(c_m * Float64(x_m * s_m))));
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(x_m * Float64(s_m * Float64(x_m * c_m))) * Float64(s_m * c_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 (x_m <= 5.2e-100)
		tmp = (1.0 / c_m) / ((x_m * s_m) * (c_m * (x_m * s_m)));
	else
		tmp = cos((x_m * -2.0)) / ((x_m * (s_m * (x_m * c_m))) * (s_m * c_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[x$95$m, 5.2e-100], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1999999999999997e-100

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. associate-/r* 59.1%

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

      2. *-commutative 59.1%

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

      3. unpow2 59.1%

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

      4. unpow2 59.1%

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

      5. swap-sqr 75.7%

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

      6. unpow2 75.7%

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

      7. associate-/r* 76.3%

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

      8. unpow2 76.3%

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

      9. unpow2 76.3%

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

      10. swap-sqr 86.8%

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

      11. unpow2 86.8%

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

      12. *-commutative 86.8%

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

   5. Simplified 86.8%

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

      1. pow-flip 86.8%

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

      2. *-commutative 86.8%

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

      3. pow-flip 86.8%

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

      4. add-sqr-sqrt 86.8%

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

      5. sqrt-div 86.8%

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

      6. metadata-eval 86.8%

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

      7. sqrt-pow1 63.6%

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

      8. associate-*r* 62.5%

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

      9. *-commutative 62.5%

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

      10. metadata-eval 62.5%

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

      11. pow1 62.5%

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

      12. sqrt-div 62.5%

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

      13. metadata-eval 62.5%

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

      14. sqrt-pow1 85.4%

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

      15. associate-*r* 86.0%

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

      16. *-commutative 86.0%

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

      17. metadata-eval 86.0%

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

      18. pow1 86.0%

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

   7. Applied egg-rr 86.0%

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

      1. un-div-inv 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*r* 85.5%

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

      4. associate-/r* 85.5%

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

      5. *-commutative 85.5%

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

      6. associate-*r* 86.9%

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

      7. associate-/l/ 86.7%

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

      8. *-commutative 86.7%

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

      9. *-commutative 86.7%

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

   9. Applied egg-rr 86.7%

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

   if 5.1999999999999997e-100 < x

   1. Initial program 71.9%

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

      1. associate-/r* 71.2%

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

      2. associate-*l* 71.2%

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

      3. unpow2 71.2%

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

      4. sqr-neg 71.2%

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

      5. unpow2 71.2%

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

      6. *-commutative 71.2%

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

      7. *-commutative 71.2%

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

      8. associate-/r* 71.9%

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

      9. cos-neg 71.9%

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

      10. *-commutative 71.9%

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

      11. distribute-rgt-neg-in 71.9%

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

      12. metadata-eval 71.9%

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

      13. associate-*r* 74.4%

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

      14. *-commutative 74.4%

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

      15. unpow2 74.4%

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

      16. sqr-neg 74.4%

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

      17. associate-*l* 78.7%

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

      18. associate-*r* 82.1%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around inf 65.8%

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

      1. associate-/r* 65.7%

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

      2. *-commutative 65.7%

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

      3. unpow2 65.7%

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

      4. unpow2 65.7%

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

      5. swap-sqr 77.7%

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

      6. unpow2 77.7%

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

      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 99.7%

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

      12. unpow2 99.7%

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

      13. associate-*r* 97.6%

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

      14. *-commutative 97.6%

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

   7. Simplified 97.6%

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

      1. unpow2 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.7%

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

      4. associate-*r* 98.7%

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

      5. associate-*l* 96.7%

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

      6. associate-*r* 95.7%

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

      7. *-commutative 95.7%

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

   9. Applied egg-rr 95.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot t_0\right)}\\ \end{array} \end{array} \]

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 5e-11)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (* x_m -2.0)) (* (* x_m c_m) (* s_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 <= 5e-11) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m * -2.0)) / ((x_m * c_m) * (s_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 <= 5d-11) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m * (-2.0d0))) / ((x_m * c_m) * (s_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 <= 5e-11) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((x_m * c_m) * (s_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 <= 5e-11:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m * -2.0)) / ((x_m * c_m) * (s_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 <= 5e-11)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(x_m * c_m) * Float64(s_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 <= 5e-11)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m * -2.0)) / ((x_m * c_m) * (s_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, 5e-11], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000018e-11

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0 62.0%

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

      1. associate-/r* 61.4%

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

      2. *-commutative 61.4%

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

      3. unpow2 61.4%

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

      4. unpow2 61.4%

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

      5. swap-sqr 76.2%

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

      6. unpow2 76.2%

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

      7. associate-/r* 76.8%

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

      8. unpow2 76.8%

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

      9. unpow2 76.8%

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

      10. swap-sqr 88.2%

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

      11. unpow2 88.2%

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

      12. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. pow-flip 88.2%

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

      2. *-commutative 88.2%

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

      3. pow-flip 88.2%

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

      4. add-sqr-sqrt 88.2%

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

      5. sqrt-div 88.2%

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

      6. metadata-eval 88.2%

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

      7. sqrt-pow1 62.7%

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

      8. associate-*r* 61.7%

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

      9. *-commutative 61.7%

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

      10. metadata-eval 61.7%

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

      11. pow1 61.7%

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

      12. sqrt-div 61.7%

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

      13. metadata-eval 61.7%

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

      14. sqrt-pow1 87.0%

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

      15. associate-*r* 87.4%

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

      16. *-commutative 87.4%

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

      17. metadata-eval 87.4%

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

      18. pow1 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. frac-times 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r* 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*r* 88.2%

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

      6. frac-times 88.2%

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

      7. frac-2neg 88.2%

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

      8. metadata-eval 88.2%

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

      9. frac-2neg 88.2%

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

      10. metadata-eval 88.2%

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

      11. frac-times 88.2%

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

      12. metadata-eval 88.2%

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

      13. distribute-rgt-neg-in 88.2%

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

      14. *-commutative 88.2%

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

      15. distribute-rgt-neg-in 88.2%

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

      16. *-commutative 88.2%

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

   9. Applied egg-rr 88.2%

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

   if 5.00000000000000018e-11 < x

   1. Initial program 69.4%

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

      1. associate-/r* 68.5%

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

      2. associate-*l* 68.5%

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

      3. unpow2 68.5%

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

      4. sqr-neg 68.5%

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

      5. unpow2 68.5%

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

      6. *-commutative 68.5%

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

      7. *-commutative 68.5%

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

      8. associate-/r* 69.4%

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

      9. cos-neg 69.4%

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

      10. *-commutative 69.4%

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

      11. distribute-rgt-neg-in 69.4%

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

      12. metadata-eval 69.4%

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

      13. associate-*r* 71.2%

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

      14. *-commutative 71.2%

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

      15. unpow2 71.2%

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

      16. sqr-neg 71.2%

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

      17. associate-*l* 76.7%

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

      18. associate-*r* 79.7%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in x around inf 61.6%

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

      1. associate-/r* 61.5%

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

      2. *-commutative 61.5%

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

      3. unpow2 61.5%

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

      4. unpow2 61.5%

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

      5. swap-sqr 76.9%

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

      6. unpow2 76.9%

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

      7. associate-/r* 77.8%

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

      8. *-commutative 77.8%

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

      9. unpow2 77.8%

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

      10. unpow2 77.8%

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

      11. swap-sqr 99.6%

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

      12. unpow2 99.6%

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

      13. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. unpow-prod-down 77.8%

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

      2. *-commutative 77.8%

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

      3. pow-prod-down 99.6%

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

      4. pow2 99.6%

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

      5. associate-*r* 97.1%

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

      6. *-commutative 97.1%

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

      7. associate-*r* 94.5%

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

      8. *-commutative 94.5%

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

   9. Applied egg-rr 94.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \frac{1}{t_0} \cdot \frac{\cos \left(x_m \cdot 2\right)}{t_0} \end{array} \end{array} \]

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)))) (* (/ 1.0 t_0) (/ (cos (* x_m 2.0)) 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);
	return (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
}

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
    t_0 = c_m * (x_m * s_m)
    code = (1.0d0 / t_0) * (cos((x_m * 2.0d0)) / t_0)
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);
	return (1.0 / t_0) * (Math.cos((x_m * 2.0)) / t_0);
}

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)
	return (1.0 / t_0) * (math.cos((x_m * 2.0)) / t_0)

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))
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x_m * 2.0)) / t_0))
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)
	t_0 = c_m * (x_m * s_m);
	tmp = (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
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]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.6%

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

   1. *-un-lft-identity 71.6%

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

   2. add-sqr-sqrt 71.6%

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

   3. times-frac 71.5%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 7: 79.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;-{t_0}^{-2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 1.15e+51) (/ 1.0 (* t_0 t_0)) (- (pow t_0 -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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.15e+51) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = -pow(t_0, -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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 1.15d+51) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = -(t_0 ** (-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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.15e+51) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = -Math.pow(t_0, -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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.15e+51:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = -math.pow(t_0, -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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.15e+51)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(-(t_0 ^ -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)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.15e+51)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = -(t_0 ^ -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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.15e+51], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], (-N[Power[t$95$0, -2.0], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000003e51

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 60.5%

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

      1. associate-/r* 60.0%

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

      2. *-commutative 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

      5. swap-sqr 73.3%

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

      6. unpow2 73.3%

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

      7. associate-/r* 73.8%

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

      8. unpow2 73.8%

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

      9. unpow2 73.8%

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

      10. swap-sqr 85.0%

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

      11. unpow2 85.0%

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

      12. *-commutative 85.0%

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

   5. Simplified 85.0%

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

      1. pow-flip 84.9%

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

      2. *-commutative 84.9%

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

      3. pow-flip 85.0%

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

      4. add-sqr-sqrt 84.9%

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

      5. sqrt-div 84.9%

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

      6. metadata-eval 84.9%

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

      7. sqrt-pow1 60.5%

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

      8. associate-*r* 59.5%

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

      9. *-commutative 59.5%

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

      10. metadata-eval 59.5%

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

      11. pow1 59.5%

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

      12. sqrt-div 59.5%

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

      13. metadata-eval 59.5%

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

      14. sqrt-pow1 83.8%

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

      15. associate-*r* 84.2%

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

      16. *-commutative 84.2%

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

      17. metadata-eval 84.2%

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

      18. pow1 84.2%

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

   7. Applied egg-rr 84.2%

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

      1. frac-times 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-*r* 83.9%

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

      4. *-commutative 83.9%

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

      5. associate-*r* 85.0%

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

      6. frac-times 84.9%

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

      7. frac-2neg 84.9%

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

      8. metadata-eval 84.9%

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

      9. frac-2neg 84.9%

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

      10. metadata-eval 84.9%

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

      11. frac-times 85.0%

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

      12. metadata-eval 85.0%

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

      13. distribute-rgt-neg-in 85.0%

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

      14. *-commutative 85.0%

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

      15. distribute-rgt-neg-in 85.0%

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

      16. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

   if 1.15000000000000003e51 < x

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0 50.3%

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

      1. associate-/r* 50.3%

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

      2. *-commutative 50.3%

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

      3. unpow2 50.3%

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

      4. unpow2 50.3%

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

      5. swap-sqr 62.7%

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

      6. unpow2 62.7%

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

      7. associate-/r* 62.7%

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

      8. unpow2 62.7%

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

      9. unpow2 62.7%

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

      10. swap-sqr 71.0%

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

      11. unpow2 71.0%

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

      12. *-commutative 71.0%

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

   5. Simplified 71.0%

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

      1. pow-flip 71.0%

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

      2. *-commutative 71.0%

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

      3. pow-flip 71.0%

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

      4. add-sqr-sqrt 71.0%

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

      5. sqrt-div 71.0%

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

      6. metadata-eval 71.0%

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

      7. sqrt-pow1 76.3%

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

      8. associate-*r* 76.1%

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

      9. *-commutative 76.1%

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

      10. metadata-eval 76.1%

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

      11. pow1 76.1%

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

      12. sqrt-div 76.1%

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

      13. metadata-eval 76.1%

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

      14. sqrt-pow1 70.8%

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

      15. associate-*r* 70.8%

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

      16. *-commutative 70.8%

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

      17. metadata-eval 70.8%

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

      18. pow1 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. un-div-inv 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r* 70.8%

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

      4. associate-/r* 70.8%

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

      5. *-commutative 70.8%

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

      6. associate-*r* 71.0%

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

      7. associate-/l/ 70.7%

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

      8. *-commutative 70.7%

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

      9. *-commutative 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. div-inv 70.7%

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

       2. frac-times 70.7%

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

       3. metadata-eval 70.7%

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

       4. *-commutative 70.7%

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

       5. add-sqr-sqrt 31.8%

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

       6. sqrt-unprod 71.1%

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

       7. sqr-neg 71.1%

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

       8. sqrt-unprod 39.9%

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

       9. add-sqr-sqrt 72.6%

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

       10. add-sqr-sqrt 32.7%

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

       11. sqrt-unprod 71.4%

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

       12. sqr-neg 71.4%

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

       13. sqrt-unprod 38.9%

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

       14. add-sqr-sqrt 70.7%

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

       15. associate-*l* 71.0%

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

       16. frac-2neg 71.0%

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

   11. Applied egg-rr 72.2%

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

       1. mul-1-neg 72.2%

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

       2. associate-*r* 72.6%

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

   13. Simplified 72.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x_m \cdot s_m\right)\\ \mathbf{if}\;x_m \leq 5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot \left(x_m \cdot \left(s_m \cdot c_m\right)\right)}\\ \end{array} \end{array} \]

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 5.5e+45)
     (/ 1.0 (* t_0 t_0))
     (/ 1.0 (* t_0 (* x_m (* s_m c_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 <= 5.5e+45) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (t_0 * (x_m * (s_m * c_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 <= 5.5d+45) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = 1.0d0 / (t_0 * (x_m * (s_m * c_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 <= 5.5e+45) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (t_0 * (x_m * (s_m * c_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 <= 5.5e+45:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = 1.0 / (t_0 * (x_m * (s_m * c_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 <= 5.5e+45)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(x_m * Float64(s_m * c_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 <= 5.5e+45)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = 1.0 / (t_0 * (x_m * (s_m * c_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, 5.5e+45], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5000000000000001e45

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0 60.9%

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

      1. associate-/r* 60.4%

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

      2. *-commutative 60.4%

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

      3. unpow2 60.4%

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

      4. unpow2 60.4%

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

      5. swap-sqr 73.9%

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

      6. unpow2 73.9%

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

      7. associate-/r* 74.5%

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

      8. unpow2 74.5%

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

      9. unpow2 74.5%

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

      10. swap-sqr 85.6%

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

      11. unpow2 85.6%

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

      12. *-commutative 85.6%

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

   5. Simplified 85.6%

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

      1. pow-flip 85.6%

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

      2. *-commutative 85.6%

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

      3. pow-flip 85.6%

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

      4. add-sqr-sqrt 85.6%

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

      5. sqrt-div 85.6%

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

      6. metadata-eval 85.6%

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

      7. sqrt-pow1 61.2%

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

      8. associate-*r* 60.2%

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

      9. *-commutative 60.2%

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

      10. metadata-eval 60.2%

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

      11. pow1 60.2%

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

      12. sqrt-div 60.2%

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

      13. metadata-eval 60.2%

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

      14. sqrt-pow1 84.5%

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

      15. associate-*r* 84.9%

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

      16. *-commutative 84.9%

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

      17. metadata-eval 84.9%

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

      18. pow1 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. frac-times 85.0%

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

      2. *-commutative 85.0%

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

      3. associate-*r* 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-*r* 85.6%

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

      6. frac-times 85.6%

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

      7. frac-2neg 85.6%

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

      8. metadata-eval 85.6%

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

      9. frac-2neg 85.6%

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

      10. metadata-eval 85.6%

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

      11. frac-times 85.6%

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

      12. metadata-eval 85.6%

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

      13. distribute-rgt-neg-in 85.6%

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

      14. *-commutative 85.6%

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

      15. distribute-rgt-neg-in 85.6%

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

      16. *-commutative 85.6%

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

   9. Applied egg-rr 85.6%

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

   if 5.5000000000000001e45 < x

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0 49.4%

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

      1. associate-/r* 49.4%

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

      2. *-commutative 49.4%

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

      3. unpow2 49.4%

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

      4. unpow2 49.4%

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

      5. swap-sqr 61.1%

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

      6. unpow2 61.1%

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

      7. associate-/r* 61.1%

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

      8. unpow2 61.1%

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

      9. unpow2 61.1%

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

      10. swap-sqr 69.3%

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

      11. unpow2 69.3%

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

      12. *-commutative 69.3%

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

   5. Simplified 69.3%

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

      1. pow-flip 69.3%

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

      2. *-commutative 69.3%

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

      3. pow-flip 69.3%

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

      4. add-sqr-sqrt 69.3%

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

      5. sqrt-div 69.3%

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

      6. metadata-eval 69.3%

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

      7. sqrt-pow1 72.9%

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

      8. associate-*r* 72.7%

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

      9. *-commutative 72.7%

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

      10. metadata-eval 72.7%

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

      11. pow1 72.7%

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

      12. sqrt-div 72.7%

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

      13. metadata-eval 72.7%

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

      14. sqrt-pow1 69.2%

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

      15. associate-*r* 69.2%

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

      16. *-commutative 69.2%

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

      17. metadata-eval 69.2%

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

      18. pow1 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. frac-times 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*r* 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-*r* 69.3%

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

      6. frac-times 69.3%

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

      7. frac-2neg 69.3%

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

      8. metadata-eval 69.3%

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

      9. frac-2neg 69.3%

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

      10. metadata-eval 69.3%

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

      11. frac-times 69.3%

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

      12. metadata-eval 69.3%

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

      13. distribute-rgt-neg-in 69.3%

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

      14. *-commutative 69.3%

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

      15. distribute-rgt-neg-in 69.3%

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

      16. *-commutative 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. distribute-rgt-neg-out 69.3%

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

       2. neg-sub0 67.5%

          \[\leadsto \frac{1}{\color{blue}{\left(0 - c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot \left(-s \cdot x\right)\right)} \]

       3. associate-*r* 65.8%

          \[\leadsto \frac{1}{\left(0 - \color{blue}{\left(c \cdot s\right) \cdot x}\right) \cdot \left(c \cdot \left(-s \cdot x\right)\right)} \]

   11. Applied egg-rr 65.8%

       \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot \left(-s \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 24.1× speedup

Math

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

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) (* (* 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 (1.0 / 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 = (1.0d0 / 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 (1.0 / 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 (1.0 / 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(Float64(1.0 / 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 = (1.0 / 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[(1.0 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around 0 58.4%

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

   1. associate-/r* 58.0%

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

   2. *-commutative 58.0%

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

   3. unpow2 58.0%

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

   4. unpow2 58.0%

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

   5. swap-sqr 71.1%

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

   6. unpow2 71.1%

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

   7. associate-/r* 71.5%

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

   8. unpow2 71.5%

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

   9. unpow2 71.5%

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

   10. swap-sqr 82.1%

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

   11. unpow2 82.1%

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

   12. *-commutative 82.1%

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

5. Simplified 82.1%

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

   1. pow-flip 82.1%

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

   2. *-commutative 82.1%

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

   3. pow-flip 82.1%

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

   4. add-sqr-sqrt 82.0%

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

   5. sqrt-div 82.0%

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

   6. metadata-eval 82.0%

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

   7. sqrt-pow1 63.8%

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

   8. associate-*r* 63.0%

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

   9. *-commutative 63.0%

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

   10. metadata-eval 63.0%

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

   11. pow1 63.0%

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

   12. sqrt-div 63.0%

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

   13. metadata-eval 63.0%

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

   14. sqrt-pow1 81.1%

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

   15. associate-*r* 81.5%

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

   16. *-commutative 81.5%

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

   17. metadata-eval 81.5%

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

   18. pow1 81.5%

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

7. Applied egg-rr 81.5%

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

   1. un-div-inv 81.5%

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

   2. *-commutative 81.5%

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

   3. associate-*r* 81.2%

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

   4. associate-/r* 81.2%

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

   5. *-commutative 81.2%

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

   6. associate-*r* 82.1%

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

   7. associate-/l/ 81.5%

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

   8. *-commutative 81.5%

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

   9. *-commutative 81.5%

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

9. Applied egg-rr 81.5%

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

10. Final simplification 81.5%

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

Alternative 10: 79.7% accurate, 24.1× speedup

Math

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

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)))) (/ (/ 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);
	return (1.0 / t_0) / t_0;
}

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
    t_0 = c_m * (x_m * s_m)
    code = (1.0d0 / t_0) / t_0
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);
	return (1.0 / t_0) / t_0;
}

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)
	return (1.0 / t_0) / t_0

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))
	return Float64(Float64(1.0 / t_0) / t_0)
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)
	t_0 = c_m * (x_m * s_m);
	tmp = (1.0 / t_0) / t_0;
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]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around 0 58.4%

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

   1. associate-/r* 58.0%

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

   2. *-commutative 58.0%

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

   3. unpow2 58.0%

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

   4. unpow2 58.0%

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

   5. swap-sqr 71.1%

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

   6. unpow2 71.1%

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

   7. associate-/r* 71.5%

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

   8. unpow2 71.5%

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

   9. unpow2 71.5%

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

   10. swap-sqr 82.1%

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

   11. unpow2 82.1%

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

   12. *-commutative 82.1%

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

5. Simplified 82.1%

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

   1. pow-flip 82.1%

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

   2. *-commutative 82.1%

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

   3. pow-flip 82.1%

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

   4. add-sqr-sqrt 82.0%

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

   5. sqrt-div 82.0%

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

   6. metadata-eval 82.0%

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

   7. sqrt-pow1 63.8%

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

   8. associate-*r* 63.0%

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

   9. *-commutative 63.0%

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

   10. metadata-eval 63.0%

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

   11. pow1 63.0%

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

   12. sqrt-div 63.0%

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

   13. metadata-eval 63.0%

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

   14. sqrt-pow1 81.1%

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

   15. associate-*r* 81.5%

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

   16. *-commutative 81.5%

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

   17. metadata-eval 81.5%

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

   18. pow1 81.5%

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

7. Applied egg-rr 81.5%

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

   1. un-div-inv 81.5%

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

   2. associate-*r* 80.8%

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

   3. *-commutative 80.8%

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

   4. associate-*r* 81.2%

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

   5. associate-*r* 81.4%

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

   6. *-commutative 81.4%

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

   7. associate-*r* 82.1%

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

9. Applied egg-rr 82.1%

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

10. Final simplification 82.1%

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

Alternative 11: 79.6% accurate, 24.1× speedup

Math

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

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)))) (/ 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);
	return 1.0 / (t_0 * t_0);
}

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
    t_0 = c_m * (x_m * s_m)
    code = 1.0d0 / (t_0 * t_0)
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);
	return 1.0 / (t_0 * t_0);
}

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)
	return 1.0 / (t_0 * t_0)

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))
	return Float64(1.0 / Float64(t_0 * t_0))
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)
	t_0 = c_m * (x_m * s_m);
	tmp = 1.0 / (t_0 * t_0);
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]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around 0 58.4%

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

   1. associate-/r* 58.0%

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

   2. *-commutative 58.0%

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

   3. unpow2 58.0%

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

   4. unpow2 58.0%

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

   5. swap-sqr 71.1%

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

   6. unpow2 71.1%

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

   7. associate-/r* 71.5%

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

   8. unpow2 71.5%

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

   9. unpow2 71.5%

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

   10. swap-sqr 82.1%

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

   11. unpow2 82.1%

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

   12. *-commutative 82.1%

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

5. Simplified 82.1%

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

   1. pow-flip 82.1%

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

   2. *-commutative 82.1%

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

   3. pow-flip 82.1%

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

   4. add-sqr-sqrt 82.0%

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

   5. sqrt-div 82.0%

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

   6. metadata-eval 82.0%

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

   7. sqrt-pow1 63.8%

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

   8. associate-*r* 63.0%

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

   9. *-commutative 63.0%

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

   10. metadata-eval 63.0%

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

   11. pow1 63.0%

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

   12. sqrt-div 63.0%

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

   13. metadata-eval 63.0%

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

   14. sqrt-pow1 81.1%

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

   15. associate-*r* 81.5%

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

   16. *-commutative 81.5%

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

   17. metadata-eval 81.5%

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

   18. pow1 81.5%

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

7. Applied egg-rr 81.5%

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

   1. frac-times 81.5%

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

   2. *-commutative 81.5%

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

   3. associate-*r* 81.2%

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

   4. *-commutative 81.2%

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

   5. associate-*r* 82.1%

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

   6. frac-times 82.0%

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

   7. frac-2neg 82.0%

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

   8. metadata-eval 82.0%

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

   9. frac-2neg 82.0%

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

   10. metadata-eval 82.0%

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

   11. frac-times 82.1%

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

   12. metadata-eval 82.1%

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

   13. distribute-rgt-neg-in 82.1%

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

   14. *-commutative 82.1%

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

   15. distribute-rgt-neg-in 82.1%

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

   16. *-commutative 82.1%

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

9. Applied egg-rr 82.1%

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

10. Final simplification 82.1%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))