mixedcos

Percentage Accurate: 65.7% → 99.0%

Time: 17.2s

Alternatives: 7

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% 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: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot -2\right)\\ t_1 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2 \cdot 10^{-25}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{elif}\;x\_m \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{t\_0}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot t\_1\right)}\\ \end{array} \end{array} \]

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))) (t_1 (* s_m (* x_m c_m))))
   (if (<= x_m 2e-25)
     (pow (* c_m (* x_m s_m)) -2.0)
     (if (<= x_m 2e+126)
       (/ t_0 (* (* s_m c_m) (* x_m t_1)))
       (/ t_0 (* (* x_m c_m) (* s_m t_1)))))))

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 t_1 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 2e-25) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else if (x_m <= 2e+126) {
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	} else {
		tmp = t_0 / ((x_m * c_m) * (s_m * t_1));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * (-2.0d0)))
    t_1 = s_m * (x_m * c_m)
    if (x_m <= 2d-25) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else if (x_m <= 2d+126) then
        tmp = t_0 / ((s_m * c_m) * (x_m * t_1))
    else
        tmp = t_0 / ((x_m * c_m) * (s_m * t_1))
    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 t_1 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 2e-25) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else if (x_m <= 2e+126) {
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	} else {
		tmp = t_0 / ((x_m * c_m) * (s_m * t_1));
	}
	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))
	t_1 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 2e-25:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	elif x_m <= 2e+126:
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1))
	else:
		tmp = t_0 / ((x_m * c_m) * (s_m * t_1))
	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))
	t_1 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 2e-25)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	elseif (x_m <= 2e+126)
		tmp = Float64(t_0 / Float64(Float64(s_m * c_m) * Float64(x_m * t_1)));
	else
		tmp = Float64(t_0 / Float64(Float64(x_m * c_m) * Float64(s_m * t_1)));
	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));
	t_1 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 2e-25)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	elseif (x_m <= 2e+126)
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	else
		tmp = t_0 / ((x_m * c_m) * (s_m * t_1));
	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]}, Block[{t$95$1 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2e-25], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x$95$m, 2e+126], N[(t$95$0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.00000000000000008e-25

   1. Initial program 70.6%

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

      1. associate-/r* 70.6%

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

      2. associate-*l* 70.6%

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

      3. unpow2 70.6%

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

      4. sqr-neg 70.6%

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

      5. unpow2 70.6%

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

      6. *-commutative 70.6%

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

      7. *-commutative 70.6%

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

      8. associate-/r* 70.6%

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

      9. cos-neg 70.6%

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

      10. *-commutative 70.6%

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

      11. distribute-rgt-neg-in 70.6%

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

      12. metadata-eval 70.6%

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

      13. associate-*r* 70.3%

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

      14. *-commutative 70.3%

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

      15. unpow2 70.3%

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

      16. sqr-neg 70.3%

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

      17. associate-*l* 75.3%

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

      18. associate-*r* 76.6%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around inf 65.2%

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

      1. associate-/r* 65.2%

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

      2. *-commutative 65.2%

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

      3. unpow2 65.2%

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

      4. unpow2 65.2%

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

      5. swap-sqr 78.2%

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

      6. unpow2 78.2%

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

      7. associate-/r* 78.2%

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

      8. *-commutative 78.2%

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

      9. unpow2 78.2%

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

      10. unpow2 78.2%

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

      11. swap-sqr 96.1%

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

      12. unpow2 96.1%

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

      13. associate-*r* 96.6%

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

      14. *-commutative 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in x around 0 57.2%

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

      1. associate-*r* 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

      4. unpow2 56.1%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. swap-sqr 80.2%

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

      8. associate-*r* 78.6%

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

      9. associate-*r* 79.2%

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

      10. /-rgt-identity 79.2%

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

      11. associate-/r/ 79.2%

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

      12. associate-/l* 79.4%

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

      13. associate-*l/ 79.4%

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

      14. unpow-1 79.4%

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

      15. unpow-1 79.4%

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

      16. pow-sqr 79.4%

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

      17. metadata-eval 79.4%

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

      18. associate-*r* 80.4%

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

      19. *-commutative 80.4%

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

      20. associate-*r* 80.8%

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

   10. Simplified 80.8%

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

   if 2.00000000000000008e-25 < x < 1.99999999999999985e126

   1. Initial program 76.0%

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

      1. associate-/r* 76.0%

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

      2. associate-*l* 76.0%

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

      3. unpow2 76.0%

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

      4. sqr-neg 76.0%

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

      5. unpow2 76.0%

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

      6. *-commutative 76.0%

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

      7. *-commutative 76.0%

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

      8. associate-/r* 76.0%

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

      9. cos-neg 76.0%

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

      10. *-commutative 76.0%

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

      11. distribute-rgt-neg-in 76.0%

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

      12. metadata-eval 76.0%

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

      13. associate-*r* 79.1%

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

      14. *-commutative 79.1%

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

      15. unpow2 79.1%

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

      16. sqr-neg 79.1%

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

      17. associate-*l* 79.0%

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

      18. associate-*r* 82.2%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in x around inf 76.0%

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

      1. associate-/r* 76.0%

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

      2. *-commutative 76.0%

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

      3. unpow2 76.0%

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

      4. unpow2 76.0%

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

      5. swap-sqr 75.8%

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

      6. unpow2 75.8%

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

      7. associate-/r* 75.8%

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

      8. *-commutative 75.8%

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

      9. unpow2 75.8%

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

      10. unpow2 75.8%

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

      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. associate-*r* 99.5%

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

      14. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.6%

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

      3. unpow2 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.7%

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

      6. associate-*l* 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r* 99.5%

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

      9. *-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 1.99999999999999985e126 < x

   1. Initial program 66.9%

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

      1. associate-/r* 67.4%

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

      2. associate-*l* 67.4%

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

      3. unpow2 67.4%

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

      4. sqr-neg 67.4%

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

      5. unpow2 67.4%

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

      6. *-commutative 67.4%

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

      7. *-commutative 67.4%

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

      8. associate-/r* 66.9%

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

      9. cos-neg 66.9%

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

      10. *-commutative 66.9%

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

      11. distribute-rgt-neg-in 66.9%

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

      12. metadata-eval 66.9%

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

      13. associate-*r* 67.0%

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

      14. *-commutative 67.0%

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

      15. unpow2 67.0%

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

      16. sqr-neg 67.0%

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

      17. associate-*l* 74.2%

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

      18. associate-*r* 75.0%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in x around inf 54.3%

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

      1. associate-/r* 54.1%

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

      2. *-commutative 54.1%

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

      3. unpow2 54.1%

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

      4. unpow2 54.1%

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

      5. swap-sqr 76.8%

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

      6. unpow2 76.8%

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

      7. associate-/r* 77.4%

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

      8. *-commutative 77.4%

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

      9. unpow2 77.4%

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

      10. unpow2 77.4%

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

      11. swap-sqr 99.0%

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

      12. unpow2 99.0%

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

      13. associate-*r* 96.9%

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

      14. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. unpow2 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*r* 96.9%

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

      4. associate-*r* 94.7%

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

      5. associate-*r* 94.7%

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

      6. *-commutative 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-25}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= x_m 1.1e-25)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* (* s_m c_m) (* x_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 tmp;
	if (x_m <= 1.1e-25) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_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) :: tmp
    if (x_m <= 1.1d-25) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / ((s_m * c_m) * (x_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 tmp;
	if (x_m <= 1.1e-25) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_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):
	tmp = 0
	if x_m <= 1.1e-25:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_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)
	tmp = 0.0
	if (x_m <= 1.1e-25)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(s_m * c_m) * Float64(x_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)
	tmp = 0.0;
	if (x_m <= 1.1e-25)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_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_] := If[LessEqual[x$95$m, 1.1e-25], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * 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 < 1.1000000000000001e-25

   1. Initial program 70.6%

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

      1. associate-/r* 70.6%

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

      2. associate-*l* 70.6%

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

      3. unpow2 70.6%

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

      4. sqr-neg 70.6%

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

      5. unpow2 70.6%

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

      6. *-commutative 70.6%

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

      7. *-commutative 70.6%

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

      8. associate-/r* 70.6%

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

      9. cos-neg 70.6%

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

      10. *-commutative 70.6%

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

      11. distribute-rgt-neg-in 70.6%

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

      12. metadata-eval 70.6%

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

      13. associate-*r* 70.3%

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

      14. *-commutative 70.3%

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

      15. unpow2 70.3%

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

      16. sqr-neg 70.3%

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

      17. associate-*l* 75.3%

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

      18. associate-*r* 76.6%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around inf 65.2%

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

      1. associate-/r* 65.2%

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

      2. *-commutative 65.2%

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

      3. unpow2 65.2%

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

      4. unpow2 65.2%

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

      5. swap-sqr 78.2%

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

      6. unpow2 78.2%

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

      7. associate-/r* 78.2%

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

      8. *-commutative 78.2%

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

      9. unpow2 78.2%

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

      10. unpow2 78.2%

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

      11. swap-sqr 96.1%

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

      12. unpow2 96.1%

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

      13. associate-*r* 96.6%

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

      14. *-commutative 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in x around 0 57.2%

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

      1. associate-*r* 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

      4. unpow2 56.1%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. swap-sqr 80.2%

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

      8. associate-*r* 78.6%

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

      9. associate-*r* 79.2%

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

      10. /-rgt-identity 79.2%

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

      11. associate-/r/ 79.2%

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

      12. associate-/l* 79.4%

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

      13. associate-*l/ 79.4%

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

      14. unpow-1 79.4%

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

      15. unpow-1 79.4%

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

      16. pow-sqr 79.4%

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

      17. metadata-eval 79.4%

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

      18. associate-*r* 80.4%

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

      19. *-commutative 80.4%

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

      20. associate-*r* 80.8%

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

   10. Simplified 80.8%

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

   if 1.1000000000000001e-25 < x

   1. Initial program 70.8%

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

      1. associate-/r* 71.1%

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

      2. associate-*l* 71.1%

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

      3. unpow2 71.1%

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

      4. sqr-neg 71.1%

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

      5. unpow2 71.1%

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

      6. *-commutative 71.1%

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

      7. *-commutative 71.1%

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

      8. associate-/r* 70.8%

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

      9. cos-neg 70.8%

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

      10. *-commutative 70.8%

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

      11. distribute-rgt-neg-in 70.8%

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

      12. metadata-eval 70.8%

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

      13. associate-*r* 72.2%

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

      14. *-commutative 72.2%

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

      15. unpow2 72.2%

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

      16. sqr-neg 72.2%

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

      17. associate-*l* 76.3%

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

      18. associate-*r* 78.1%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around inf 63.7%

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

      1. associate-/r* 63.6%

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

      2. *-commutative 63.6%

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

      3. unpow2 63.6%

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

      4. unpow2 63.6%

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

      5. swap-sqr 76.4%

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

      6. unpow2 76.4%

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

      7. associate-/r* 76.7%

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

      8. *-commutative 76.7%

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

      9. unpow2 76.7%

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

      10. unpow2 76.7%

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

      11. swap-sqr 99.3%

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

      12. unpow2 99.3%

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

      13. associate-*r* 98.0%

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

      14. *-commutative 98.0%

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

   7. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r* 99.3%

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

      3. unpow2 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-*r* 95.7%

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

      6. associate-*l* 95.6%

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

      7. *-commutative 95.6%

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

      8. associate-*r* 94.3%

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

      9. *-commutative 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= x_m 2.9e-25)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* 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 tmp;
	if (x_m <= 2.9e-25) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / (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) :: tmp
    if (x_m <= 2.9d-25) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / (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 tmp;
	if (x_m <= 2.9e-25) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / (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):
	tmp = 0
	if x_m <= 2.9e-25:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / (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)
	tmp = 0.0
	if (x_m <= 2.9e-25)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / 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)
	tmp = 0.0;
	if (x_m <= 2.9e-25)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / (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_] := If[LessEqual[x$95$m, 2.9e-25], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(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 < 2.9000000000000001e-25

   1. Initial program 70.6%

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

      1. associate-/r* 70.6%

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

      2. associate-*l* 70.6%

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

      3. unpow2 70.6%

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

      4. sqr-neg 70.6%

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

      5. unpow2 70.6%

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

      6. *-commutative 70.6%

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

      7. *-commutative 70.6%

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

      8. associate-/r* 70.6%

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

      9. cos-neg 70.6%

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

      10. *-commutative 70.6%

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

      11. distribute-rgt-neg-in 70.6%

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

      12. metadata-eval 70.6%

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

      13. associate-*r* 70.3%

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

      14. *-commutative 70.3%

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

      15. unpow2 70.3%

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

      16. sqr-neg 70.3%

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

      17. associate-*l* 75.3%

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

      18. associate-*r* 76.6%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around inf 65.2%

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

      1. associate-/r* 65.2%

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

      2. *-commutative 65.2%

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

      3. unpow2 65.2%

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

      4. unpow2 65.2%

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

      5. swap-sqr 78.2%

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

      6. unpow2 78.2%

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

      7. associate-/r* 78.2%

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

      8. *-commutative 78.2%

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

      9. unpow2 78.2%

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

      10. unpow2 78.2%

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

      11. swap-sqr 96.1%

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

      12. unpow2 96.1%

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

      13. associate-*r* 96.6%

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

      14. *-commutative 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in x around 0 57.2%

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

      1. associate-*r* 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

      4. unpow2 56.1%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. swap-sqr 80.2%

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

      8. associate-*r* 78.6%

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

      9. associate-*r* 79.2%

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

      10. /-rgt-identity 79.2%

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

      11. associate-/r/ 79.2%

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

      12. associate-/l* 79.4%

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

      13. associate-*l/ 79.4%

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

      14. unpow-1 79.4%

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

      15. unpow-1 79.4%

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

      16. pow-sqr 79.4%

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

      17. metadata-eval 79.4%

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

      18. associate-*r* 80.4%

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

      19. *-commutative 80.4%

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

      20. associate-*r* 80.8%

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

   10. Simplified 80.8%

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

   if 2.9000000000000001e-25 < x

   1. Initial program 70.8%

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

      1. associate-/r* 71.1%

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

      2. associate-*l* 71.1%

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

      3. unpow2 71.1%

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

      4. sqr-neg 71.1%

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

      5. unpow2 71.1%

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

      6. *-commutative 71.1%

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

      7. *-commutative 71.1%

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

      8. associate-/r* 70.8%

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

      9. cos-neg 70.8%

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

      10. *-commutative 70.8%

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

      11. distribute-rgt-neg-in 70.8%

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

      12. metadata-eval 70.8%

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

      13. associate-*r* 72.2%

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

      14. *-commutative 72.2%

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

      15. unpow2 72.2%

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

      16. sqr-neg 72.2%

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

      17. associate-*l* 76.3%

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

      18. associate-*r* 78.1%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around inf 63.7%

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

      1. associate-/r* 63.6%

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

      2. *-commutative 63.6%

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

      3. unpow2 63.6%

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

      4. unpow2 63.6%

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

      5. swap-sqr 76.4%

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

      6. unpow2 76.4%

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

      7. associate-/r* 76.7%

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

      8. *-commutative 76.7%

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

      9. unpow2 76.7%

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

      10. unpow2 76.7%

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

      11. swap-sqr 99.3%

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

      12. unpow2 99.3%

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

      13. associate-*r* 98.0%

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

      14. *-commutative 98.0%

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

   7. Simplified 98.0%

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

      1. unpow2 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r* 94.5%

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

      6. associate-*r* 94.4%

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

      7. *-commutative 94.4%

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

   9. Applied egg-rr 94.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \frac{\frac{\cos \left(x\_m \cdot -2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

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 (* s_m (* x_m c_m)))) (/ (/ (cos (* x_m -2.0)) 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 = s_m * (x_m * c_m);
	return (cos((x_m * -2.0)) / 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 = s_m * (x_m * c_m)
    code = (cos((x_m * (-2.0d0))) / 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 = s_m * (x_m * c_m);
	return (Math.cos((x_m * -2.0)) / 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 = s_m * (x_m * c_m)
	return (math.cos((x_m * -2.0)) / 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(s_m * Float64(x_m * c_m))
	return Float64(Float64(cos(Float64(x_m * -2.0)) / 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 = s_m * (x_m * c_m);
	tmp = (cos((x_m * -2.0)) / 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[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 70.7%

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

   1. *-un-lft-identity 70.7%

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

   2. associate-*r* 70.9%

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

   3. times-frac 71.0%

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

   4. *-commutative 71.0%

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

   5. associate-*r* 68.8%

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

   6. pow-prod-down 86.5%

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

4. Applied egg-rr 86.5%

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

   1. frac-times 85.9%

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

   2. *-un-lft-identity 85.9%

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

   3. add-sqr-sqrt 63.0%

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

   4. add-sqr-sqrt 85.9%

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

   5. add-sqr-sqrt 39.6%

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

   6. sqrt-unprod 67.3%

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

   7. *-commutative 67.3%

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

   8. *-commutative 67.3%

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

   9. swap-sqr 67.3%

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

   10. metadata-eval 67.3%

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

   11. metadata-eval 67.3%

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

   12. swap-sqr 67.3%

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

   13. sqrt-unprod 38.2%

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

   14. add-sqr-sqrt 85.9%

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

   15. associate-*l* 79.4%

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

   16. unpow2 79.4%

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

   17. swap-sqr 97.1%

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

   18. associate-*r* 95.3%

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

   19. associate-*r* 97.0%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 5: 80.1% accurate, 3.0× speedup

Math

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

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 (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 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 = (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.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.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(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 = (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[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 70.7%

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

   1. associate-/r* 70.8%

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

   2. associate-*l* 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. *-commutative 70.8%

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

   7. *-commutative 70.8%

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

   8. associate-/r* 70.7%

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

   9. cos-neg 70.7%

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

   10. *-commutative 70.7%

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

   11. distribute-rgt-neg-in 70.7%

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

   12. metadata-eval 70.7%

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

   13. associate-*r* 70.9%

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

   14. *-commutative 70.9%

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

   15. unpow2 70.9%

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

   16. sqr-neg 70.9%

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

   17. associate-*l* 75.6%

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

   18. associate-*r* 77.0%

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

3. Simplified 64.7%

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

5. Taylor expanded in x around inf 64.7%

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

   1. associate-/r* 64.7%

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

   2. *-commutative 64.7%

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

   3. unpow2 64.7%

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

   4. unpow2 64.7%

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

   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* 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 97.0%

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

   12. unpow2 97.0%

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

   13. associate-*r* 97.0%

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

   14. *-commutative 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in x around 0 55.8%

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

   1. associate-*r* 55.0%

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

   2. *-commutative 55.0%

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

   3. unpow2 55.0%

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

   4. unpow2 55.0%

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

   5. swap-sqr 65.3%

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

   6. unpow2 65.3%

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

   7. swap-sqr 75.6%

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

   8. associate-*r* 74.5%

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

   9. associate-*r* 74.9%

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

   10. /-rgt-identity 74.9%

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

   11. associate-/r/ 74.9%

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

   12. associate-/l* 75.1%

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

   13. associate-*l/ 75.1%

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

   14. unpow-1 75.1%

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

   15. unpow-1 75.1%

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

   16. pow-sqr 75.1%

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

   17. metadata-eval 75.1%

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

   18. associate-*r* 75.7%

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

   19. *-commutative 75.7%

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

   20. associate-*r* 76.1%

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

10. Simplified 76.1%

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

11. Final simplification 76.1%

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

Alternative 6: 80.0% accurate, 20.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ t\_0 \cdot t\_0 \end{array} \end{array} \]

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 (/ 1.0 (* c_m (* x_m s_m))))) (* 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 = 1.0 / (c_m * (x_m * s_m));
	return 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 = 1.0d0 / (c_m * (x_m * s_m))
    code = 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 = 1.0 / (c_m * (x_m * s_m));
	return 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 = 1.0 / (c_m * (x_m * s_m))
	return 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(1.0 / Float64(c_m * Float64(x_m * s_m)))
	return 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 = 1.0 / (c_m * (x_m * s_m));
	tmp = 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[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around 0 55.8%

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

   1. associate-/r* 55.7%

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

   2. *-commutative 55.7%

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

   3. unpow2 55.7%

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

   4. unpow2 55.7%

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

   5. swap-sqr 65.5%

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

   6. unpow2 65.5%

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

   7. associate-/r* 65.6%

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

   8. unpow2 65.6%

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

   9. unpow2 65.6%

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

   10. swap-sqr 75.9%

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

   11. unpow2 75.9%

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

   12. *-commutative 75.9%

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

5. Simplified 75.9%

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

   1. unpow-prod-down 65.6%

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

   2. *-commutative 65.6%

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

   3. unpow-prod-down 75.9%

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

   4. add-sqr-sqrt 75.8%

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

   5. sqrt-div 75.9%

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

   6. metadata-eval 75.9%

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

   7. sqrt-pow1 53.0%

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

   8. metadata-eval 53.0%

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

   9. associate-*r* 53.0%

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

   10. *-commutative 53.0%

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

   11. pow1 53.0%

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

   12. sqrt-div 53.0%

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

   13. metadata-eval 53.0%

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

   14. sqrt-pow1 74.9%

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

   15. metadata-eval 74.9%

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

   16. associate-*r* 75.1%

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

   17. *-commutative 75.1%

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

   18. pow1 75.1%

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

7. Applied egg-rr 75.1%

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

   1. associate-/r* 75.1%

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

   2. frac-times 74.0%

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

   3. *-un-lft-identity 74.0%

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

   4. *-commutative 74.0%

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

   5. *-commutative 74.0%

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

9. Applied egg-rr 74.0%

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

    1. *-un-lft-identity 74.0%

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

    2. frac-times 75.1%

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

    3. associate-*r* 74.9%

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

    4. *-commutative 74.9%

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

    5. *-commutative 74.9%

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

    6. associate-/l/ 74.9%

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

    7. *-commutative 74.9%

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

    8. associate-*r* 76.0%

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

11. Applied egg-rr 76.0%

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

12. Final simplification 76.0%

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

Alternative 7: 79.9% 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 70.7%

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

3. Taylor expanded in x around 0 55.8%

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

   1. associate-/r* 55.7%

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

   2. *-commutative 55.7%

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

   3. unpow2 55.7%

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

   4. unpow2 55.7%

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

   5. swap-sqr 65.5%

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

   6. unpow2 65.5%

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

   7. associate-/r* 65.6%

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

   8. unpow2 65.6%

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

   9. unpow2 65.6%

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

   10. swap-sqr 75.9%

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

   11. unpow2 75.9%

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

   12. *-commutative 75.9%

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

5. Simplified 75.9%

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

   1. unpow-prod-down 65.6%

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

   2. *-commutative 65.6%

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

   3. unpow-prod-down 75.9%

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

   4. add-sqr-sqrt 75.8%

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

   5. sqrt-div 75.9%

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

   6. metadata-eval 75.9%

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

   7. sqrt-pow1 53.0%

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

   8. metadata-eval 53.0%

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

   9. associate-*r* 53.0%

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

   10. *-commutative 53.0%

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

   11. pow1 53.0%

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

   12. sqrt-div 53.0%

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

   13. metadata-eval 53.0%

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

   14. sqrt-pow1 74.9%

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

   15. metadata-eval 74.9%

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

   16. associate-*r* 75.1%

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

   17. *-commutative 75.1%

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

   18. pow1 75.1%

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

7. Applied egg-rr 75.1%

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

   1. un-div-inv 75.1%

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

   2. clear-num 74.9%

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

   3. *-commutative 74.9%

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

   4. associate-/r* 74.9%

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

   5. *-commutative 74.9%

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

9. Applied egg-rr 74.9%

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

    1. div-inv 74.9%

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

    2. associate-*r* 74.8%

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

    3. *-commutative 74.8%

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

    4. *-commutative 74.8%

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

    5. associate-/l/ 74.8%

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

    6. *-commutative 74.8%

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

    7. inv-pow 74.8%

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

    8. pow-flip 74.8%

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

    9. associate-*r* 75.9%

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

    10. *-commutative 75.9%

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

    11. *-commutative 75.9%

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

    12. metadata-eval 75.9%

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

    13. pow1 75.9%

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

11. Applied egg-rr 75.9%

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

12. Final simplification 75.9%

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

Reproduce

herbie shell --seed 2024026 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))