mixedcos

Percentage Accurate: 66.5% → 98.3%

Time: 13.8s

Alternatives: 9

Speedup: 24.1×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x\_m \cdot 2\right) \cdot {\left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.4e-184)
   (/ (/ (/ 1.0 c_m) (* x_m s_m)) (* c_m (* x_m s_m)))
   (* (cos (* x_m 2.0)) (pow (* s_m (* x_m c_m)) -2.0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.4e-184) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = cos((x_m * 2.0)) * pow((s_m * (x_m * c_m)), -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 3.4d-184) then
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
    else
        tmp = cos((x_m * 2.0d0)) * ((s_m * (x_m * c_m)) ** (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.4e-184) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = Math.cos((x_m * 2.0)) * Math.pow((s_m * (x_m * c_m)), -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 3.4e-184:
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
	else:
		tmp = math.cos((x_m * 2.0)) * math.pow((s_m * (x_m * c_m)), -2.0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.4e-184)
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) / Float64(c_m * Float64(x_m * s_m)));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) * (Float64(s_m * Float64(x_m * c_m)) ^ -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 3.4e-184)
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	else
		tmp = cos((x_m * 2.0)) * ((s_m * (x_m * c_m)) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.4e-184], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000004e-184

   1. Initial program 68.0%

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

      1. associate-/l/ 68.0%

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

      2. remove-double-neg 68.0%

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

      3. distribute-frac-neg 68.0%

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

      4. distribute-neg-frac 68.0%

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

      5. remove-double-neg 68.0%

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

      6. *-commutative 68.0%

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

      7. associate-*r* 62.4%

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

      8. unpow2 62.4%

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

      9. associate-/r* 62.0%

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

      10. cos-neg 62.0%

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

      11. *-commutative 62.0%

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

      12. distribute-rgt-neg-in 62.0%

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

      13. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

      1. associate-/l/ 62.4%

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

      2. add-sqr-sqrt 52.1%

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

      3. sqrt-unprod 53.8%

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

      4. *-commutative 53.8%

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

      5. *-commutative 53.8%

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

      6. swap-sqr 53.8%

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

      7. metadata-eval 53.8%

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

      8. metadata-eval 53.8%

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

      9. swap-sqr 53.8%

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

      10. sqrt-unprod 6.6%

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

      11. add-sqr-sqrt 62.4%

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

      12. unpow2 62.4%

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

      13. associate-*r* 68.0%

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

      14. *-commutative 68.0%

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

      15. associate-/r* 68.0%

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

      16. *-un-lft-identity 68.0%

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

   6. Applied egg-rr 86.3%

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

      1. associate-*l/ 86.4%

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

      2. *-lft-identity 86.4%

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

      3. *-commutative 86.4%

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

      4. unpow2 86.4%

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

      5. rem-sqrt-square 86.4%

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

      6. *-commutative 86.4%

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

      7. unpow2 86.4%

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

      8. rem-sqrt-square 96.6%

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

      9. *-commutative 96.6%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in x around 0 79.1%

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

       1. associate-/r* 79.1%

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

       2. *-commutative 79.1%

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

       3. rem-square-sqrt 50.3%

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

       4. fabs-sqr 50.3%

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

       5. rem-square-sqrt 56.7%

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

       6. *-commutative 56.7%

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

   11. Simplified 56.7%

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

       1. add-sqr-sqrt 43.8%

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

       2. fabs-sqr 43.8%

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

       3. add-sqr-sqrt 79.1%

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

   13. Applied egg-rr 79.1%

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

   if 3.40000000000000004e-184 < x

   1. Initial program 69.3%

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

      1. associate-/l/ 69.3%

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

      2. remove-double-neg 69.3%

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

      3. distribute-frac-neg 69.3%

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

      4. distribute-neg-frac 69.3%

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

      5. remove-double-neg 69.3%

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

      6. *-commutative 69.3%

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

      7. associate-*r* 62.4%

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

      8. unpow2 62.4%

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

      9. associate-/r* 62.4%

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

      10. cos-neg 62.4%

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

      11. *-commutative 62.4%

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

      12. distribute-rgt-neg-in 62.4%

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

      13. metadata-eval 62.4%

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

   3. Simplified 62.4%

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

      1. associate-/l/ 62.4%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 45.5%

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

      4. *-commutative 45.5%

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

      5. *-commutative 45.5%

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

      6. swap-sqr 45.5%

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

      7. metadata-eval 45.5%

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

      8. metadata-eval 45.5%

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

      9. swap-sqr 45.5%

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

      10. sqrt-unprod 59.4%

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

      11. add-sqr-sqrt 62.4%

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

      12. unpow2 62.4%

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

      13. associate-*r* 69.3%

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

      14. *-commutative 69.3%

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

      15. associate-/r* 69.3%

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

      16. *-un-lft-identity 69.3%

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

   6. Applied egg-rr 88.6%

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

      1. associate-*l/ 88.6%

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

      2. *-lft-identity 88.6%

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

      3. *-commutative 88.6%

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

      4. unpow2 88.6%

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

      5. rem-sqrt-square 88.6%

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

      6. *-commutative 88.6%

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

      7. unpow2 88.6%

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

      8. rem-sqrt-square 98.9%

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

      9. *-commutative 98.9%

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

   8. Simplified 98.9%

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

      1. frac-2neg 98.9%

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

      2. distribute-frac-neg2 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-neg-frac2 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. add-sqr-sqrt 49.4%

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

      8. fabs-sqr 49.4%

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

      9. add-sqr-sqrt 72.5%

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

      10. distribute-rgt-neg-in 72.5%

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

      11. add-sqr-sqrt 49.4%

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

      12. fabs-sqr 49.4%

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

      13. add-sqr-sqrt 98.9%

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

   10. Applied egg-rr 98.9%

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

   12. Applied egg-rr 97.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 1.5 \cdot 10^{-235}:\\ \;\;\;\;t\_1 \cdot {\left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= c_m 1.5e-235)
     (* t_1 (pow (* x_m (* c_m s_m)) -2.0))
     (/ (/ t_1 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (c_m <= 1.5e-235) {
		tmp = t_1 * pow((x_m * (c_m * s_m)), -2.0);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    if (c_m <= 1.5d-235) then
        tmp = t_1 * ((x_m * (c_m * s_m)) ** (-2.0d0))
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (c_m <= 1.5e-235) {
		tmp = t_1 * Math.pow((x_m * (c_m * s_m)), -2.0);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if c_m <= 1.5e-235:
		tmp = t_1 * math.pow((x_m * (c_m * s_m)), -2.0)
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (c_m <= 1.5e-235)
		tmp = Float64(t_1 * (Float64(x_m * Float64(c_m * s_m)) ^ -2.0));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (c_m <= 1.5e-235)
		tmp = t_1 * ((x_m * (c_m * s_m)) ^ -2.0);
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 1.5e-235], N[(t$95$1 * N[Power[N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.4999999999999999e-235

   1. Initial program 70.5%

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

      1. associate-/l/ 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-frac-neg 70.5%

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

      4. distribute-neg-frac 70.5%

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

      5. remove-double-neg 70.5%

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

      6. *-commutative 70.5%

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

      7. associate-*r* 62.6%

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

      8. unpow2 62.6%

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

      9. associate-/r* 62.1%

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

      10. cos-neg 62.1%

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

      11. *-commutative 62.1%

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

      12. distribute-rgt-neg-in 62.1%

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

      13. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 61.4%

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

      2. div-inv 61.4%

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

      3. associate-/l* 61.4%

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

      4. add-sqr-sqrt 29.5%

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

      5. sqrt-unprod 50.7%

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

      6. *-commutative 50.7%

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

      7. *-commutative 50.7%

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

      8. swap-sqr 50.7%

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

      9. metadata-eval 50.7%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

      10. metadata-eval 50.7%

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

      11. swap-sqr 50.7%

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

      12. sqrt-unprod 28.6%

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

      13. add-sqr-sqrt 61.4%

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

      14. pow-flip 61.6%

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

      15. metadata-eval 61.6%

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

      16. pow-prod-down 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

   8. Simplified 76.3%

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

      1. unpow2 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

   10. Applied egg-rr 73.2%

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

   11. Taylor expanded in x around inf 62.6%

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

       1. associate-/r* 61.6%

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

       2. *-commutative 61.6%

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

       3. *-commutative 61.6%

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

       4. unpow2 61.6%

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

       5. unpow2 61.6%

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

       6. swap-sqr 76.6%

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

       7. unpow2 76.6%

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

       8. associate-/r* 77.6%

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

       9. unpow2 77.6%

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

       10. unpow2 77.6%

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

       11. swap-sqr 96.3%

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

       12. associate-*l* 93.9%

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

       13. associate-*l* 96.9%

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

       14. associate-/l/ 97.4%

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

       15. *-lft-identity 97.4%

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

       16. associate-*l/ 97.3%

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

       17. *-commutative 97.3%

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

   13. Simplified 97.1%

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

   if 1.4999999999999999e-235 < c

   1. Initial program 65.6%

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

      1. associate-/l/ 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-frac-neg 65.6%

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

      4. distribute-neg-frac 65.6%

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

      5. remove-double-neg 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-*r* 62.2%

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

      8. unpow2 62.2%

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

      9. associate-/r* 62.2%

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

      10. cos-neg 62.2%

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

      11. *-commutative 62.2%

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

      12. distribute-rgt-neg-in 62.2%

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

      13. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

      1. associate-/l/ 62.2%

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

      2. add-sqr-sqrt 33.0%

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

      3. sqrt-unprod 48.4%

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

      4. *-commutative 48.4%

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

      5. *-commutative 48.4%

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

      6. swap-sqr 48.4%

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

      7. metadata-eval 48.4%

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

      8. metadata-eval 48.4%

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

      9. swap-sqr 48.4%

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

      10. sqrt-unprod 25.6%

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

      11. add-sqr-sqrt 62.2%

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

      12. unpow2 62.2%

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

      13. associate-*r* 65.6%

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

      14. *-commutative 65.6%

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

      15. associate-/r* 65.6%

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

      16. *-un-lft-identity 65.6%

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

   6. Applied egg-rr 89.3%

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

      1. associate-*l/ 89.4%

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

      2. *-lft-identity 89.4%

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

      3. *-commutative 89.4%

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

      4. unpow2 89.4%

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

      5. rem-sqrt-square 89.4%

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

      6. *-commutative 89.4%

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

      7. unpow2 89.4%

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

      8. rem-sqrt-square 98.6%

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

      9. *-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. distribute-frac-neg2 98.6%

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

      3. *-commutative 98.6%

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

      4. distribute-neg-frac2 98.6%

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

      5. *-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. add-sqr-sqrt 59.6%

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

      8. fabs-sqr 59.6%

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

      9. add-sqr-sqrt 73.7%

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

      10. distribute-rgt-neg-in 73.7%

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

      11. add-sqr-sqrt 58.6%

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

      12. fabs-sqr 58.6%

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

      13. add-sqr-sqrt 98.6%

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

   10. Applied egg-rr 98.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.5 \cdot 10^{-235}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 7 \cdot 10^{-236}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\frac{1}{x\_m \cdot \left(c\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= c_m 7e-236)
     (* t_1 (/ (/ (/ 1.0 (* x_m (* c_m s_m))) (* c_m s_m)) x_m))
     (/ (/ t_1 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (c_m <= 7e-236) {
		tmp = t_1 * (((1.0 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    if (c_m <= 7d-236) then
        tmp = t_1 * (((1.0d0 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m)
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (c_m <= 7e-236) {
		tmp = t_1 * (((1.0 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if c_m <= 7e-236:
		tmp = t_1 * (((1.0 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m)
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (c_m <= 7e-236)
		tmp = Float64(t_1 * Float64(Float64(Float64(1.0 / Float64(x_m * Float64(c_m * s_m))) / Float64(c_m * s_m)) / x_m));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (c_m <= 7e-236)
		tmp = t_1 * (((1.0 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m);
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 7e-236], N[(t$95$1 * N[(N[(N[(1.0 / N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 6.99999999999999988e-236

   1. Initial program 70.5%

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

      1. associate-/l/ 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-frac-neg 70.5%

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

      4. distribute-neg-frac 70.5%

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

      5. remove-double-neg 70.5%

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

      6. *-commutative 70.5%

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

      7. associate-*r* 62.6%

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

      8. unpow2 62.6%

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

      9. associate-/r* 62.1%

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

      10. cos-neg 62.1%

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

      11. *-commutative 62.1%

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

      12. distribute-rgt-neg-in 62.1%

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

      13. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 61.4%

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

      2. div-inv 61.4%

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

      3. associate-/l* 61.4%

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

      4. add-sqr-sqrt 29.5%

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

      5. sqrt-unprod 50.7%

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

      6. *-commutative 50.7%

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

      7. *-commutative 50.7%

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

      8. swap-sqr 50.7%

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

      9. metadata-eval 50.7%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

      10. metadata-eval 50.7%

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

      11. swap-sqr 50.7%

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

      12. sqrt-unprod 28.6%

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

      13. add-sqr-sqrt 61.4%

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

      14. pow-flip 61.6%

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

      15. metadata-eval 61.6%

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

      16. pow-prod-down 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

   8. Simplified 76.3%

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

      1. unpow2 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

   10. Applied egg-rr 73.2%

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

   11. Taylor expanded in x around inf 62.6%

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

       1. associate-/r* 61.6%

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

       2. *-commutative 61.6%

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

       3. *-commutative 61.6%

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

       4. unpow2 61.6%

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

       5. unpow2 61.6%

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

       6. swap-sqr 76.6%

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

       7. unpow2 76.6%

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

       8. associate-/r* 77.6%

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

       9. unpow2 77.6%

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

       10. unpow2 77.6%

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

       11. swap-sqr 96.3%

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

       12. associate-*l* 93.9%

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

       13. associate-*l* 96.9%

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

       14. associate-/l/ 97.4%

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

       15. *-lft-identity 97.4%

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

       16. associate-*l/ 97.3%

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

       17. *-commutative 97.3%

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

   13. Simplified 97.1%

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

       1. associate-*r* 97.4%

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

       2. *-commutative 97.4%

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

       3. metadata-eval 97.4%

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

       4. pow-sqr 97.3%

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

       5. inv-pow 97.3%

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

       6. inv-pow 97.3%

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

       7. associate-*r* 94.3%

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

       8. *-commutative 94.3%

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

       9. associate-/r* 94.3%

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

       10. div-inv 94.4%

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

       11. *-commutative 94.4%

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

       12. associate-*r* 94.1%

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

       13. *-commutative 94.1%

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

       14. associate-/r* 93.2%

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

       15. associate-/r* 96.2%

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

       16. associate-/l/ 96.1%

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

       17. *-commutative 96.1%

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

       18. associate-/r* 96.2%

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

       19. *-commutative 96.2%

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

   15. Applied egg-rr 96.2%

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

   if 6.99999999999999988e-236 < c

   1. Initial program 65.6%

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

      1. associate-/l/ 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-frac-neg 65.6%

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

      4. distribute-neg-frac 65.6%

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

      5. remove-double-neg 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-*r* 62.2%

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

      8. unpow2 62.2%

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

      9. associate-/r* 62.2%

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

      10. cos-neg 62.2%

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

      11. *-commutative 62.2%

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

      12. distribute-rgt-neg-in 62.2%

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

      13. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

      1. associate-/l/ 62.2%

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

      2. add-sqr-sqrt 33.0%

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

      3. sqrt-unprod 48.4%

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

      4. *-commutative 48.4%

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

      5. *-commutative 48.4%

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

      6. swap-sqr 48.4%

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

      7. metadata-eval 48.4%

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

      8. metadata-eval 48.4%

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

      9. swap-sqr 48.4%

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

      10. sqrt-unprod 25.6%

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

      11. add-sqr-sqrt 62.2%

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

      12. unpow2 62.2%

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

      13. associate-*r* 65.6%

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

      14. *-commutative 65.6%

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

      15. associate-/r* 65.6%

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

      16. *-un-lft-identity 65.6%

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

   6. Applied egg-rr 89.3%

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

      1. associate-*l/ 89.4%

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

      2. *-lft-identity 89.4%

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

      3. *-commutative 89.4%

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

      4. unpow2 89.4%

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

      5. rem-sqrt-square 89.4%

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

      6. *-commutative 89.4%

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

      7. unpow2 89.4%

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

      8. rem-sqrt-square 98.6%

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

      9. *-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. distribute-frac-neg2 98.6%

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

      3. *-commutative 98.6%

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

      4. distribute-neg-frac2 98.6%

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

      5. *-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. add-sqr-sqrt 59.6%

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

      8. fabs-sqr 59.6%

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

      9. add-sqr-sqrt 73.7%

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

      10. distribute-rgt-neg-in 73.7%

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

      11. add-sqr-sqrt 58.6%

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

      12. fabs-sqr 58.6%

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

      13. add-sqr-sqrt 98.6%

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

   10. Applied egg-rr 98.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{-236}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot \frac{\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ t_2 := x\_m \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 6.2 \cdot 10^{-236}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{1}{t\_2}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m)))
        (t_1 (cos (* x_m 2.0)))
        (t_2 (* x_m (* c_m s_m))))
   (if (<= c_m 6.2e-236) (* t_1 (/ (/ 1.0 t_2) t_2)) (/ (/ t_1 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double t_2 = x_m * (c_m * s_m);
	double tmp;
	if (c_m <= 6.2e-236) {
		tmp = t_1 * ((1.0 / t_2) / t_2);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    t_2 = x_m * (c_m * s_m)
    if (c_m <= 6.2d-236) then
        tmp = t_1 * ((1.0d0 / t_2) / t_2)
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double t_2 = x_m * (c_m * s_m);
	double tmp;
	if (c_m <= 6.2e-236) {
		tmp = t_1 * ((1.0 / t_2) / t_2);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	t_2 = x_m * (c_m * s_m)
	tmp = 0
	if c_m <= 6.2e-236:
		tmp = t_1 * ((1.0 / t_2) / t_2)
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	t_2 = Float64(x_m * Float64(c_m * s_m))
	tmp = 0.0
	if (c_m <= 6.2e-236)
		tmp = Float64(t_1 * Float64(Float64(1.0 / t_2) / t_2));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	t_1 = cos((x_m * 2.0));
	t_2 = x_m * (c_m * s_m);
	tmp = 0.0;
	if (c_m <= 6.2e-236)
		tmp = t_1 * ((1.0 / t_2) / t_2);
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 6.2e-236], N[(t$95$1 * N[(N[(1.0 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 6.1999999999999997e-236

   1. Initial program 70.5%

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

      1. associate-/l/ 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-frac-neg 70.5%

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

      4. distribute-neg-frac 70.5%

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

      5. remove-double-neg 70.5%

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

      6. *-commutative 70.5%

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

      7. associate-*r* 62.6%

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

      8. unpow2 62.6%

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

      9. associate-/r* 62.1%

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

      10. cos-neg 62.1%

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

      11. *-commutative 62.1%

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

      12. distribute-rgt-neg-in 62.1%

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

      13. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 61.4%

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

      2. div-inv 61.4%

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

      3. associate-/l* 61.4%

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

      4. add-sqr-sqrt 29.5%

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

      5. sqrt-unprod 50.7%

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

      6. *-commutative 50.7%

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

      7. *-commutative 50.7%

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

      8. swap-sqr 50.7%

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

      9. metadata-eval 50.7%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

      10. metadata-eval 50.7%

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

      11. swap-sqr 50.7%

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

      12. sqrt-unprod 28.6%

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

      13. add-sqr-sqrt 61.4%

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

      14. pow-flip 61.6%

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

      15. metadata-eval 61.6%

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

      16. pow-prod-down 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. *-commutative 76.3%

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

   8. Simplified 76.3%

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

      1. unpow2 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 73.2%

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

      4. *-commutative 73.2%

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

   10. Applied egg-rr 73.2%

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

   11. Taylor expanded in x around inf 62.6%

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

       1. associate-/r* 61.6%

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

       2. *-commutative 61.6%

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

       3. *-commutative 61.6%

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

       4. unpow2 61.6%

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

       5. unpow2 61.6%

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

       6. swap-sqr 76.6%

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

       7. unpow2 76.6%

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

       8. associate-/r* 77.6%

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

       9. unpow2 77.6%

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

       10. unpow2 77.6%

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

       11. swap-sqr 96.3%

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

       12. associate-*l* 93.9%

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

       13. associate-*l* 96.9%

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

       14. associate-/l/ 97.4%

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

       15. *-lft-identity 97.4%

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

       16. associate-*l/ 97.3%

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

       17. *-commutative 97.3%

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

   13. Simplified 97.1%

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

       1. associate-*r* 97.4%

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

       2. *-commutative 97.4%

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

       3. metadata-eval 97.4%

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

       4. pow-sqr 97.3%

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

       5. inv-pow 97.3%

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

       6. inv-pow 97.3%

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

       7. frac-times 96.9%

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

       8. metadata-eval 96.9%

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

       9. associate-/r* 97.4%

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

       10. *-commutative 97.4%

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

       11. *-commutative 97.4%

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

       12. *-commutative 97.4%

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

       13. associate-*r* 94.8%

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

       14. *-commutative 94.8%

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

       15. associate-*r* 97.1%

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

   15. Applied egg-rr 97.1%

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

   if 6.1999999999999997e-236 < c

   1. Initial program 65.6%

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

      1. associate-/l/ 65.6%

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

      2. remove-double-neg 65.6%

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

      3. distribute-frac-neg 65.6%

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

      4. distribute-neg-frac 65.6%

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

      5. remove-double-neg 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-*r* 62.2%

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

      8. unpow2 62.2%

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

      9. associate-/r* 62.2%

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

      10. cos-neg 62.2%

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

      11. *-commutative 62.2%

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

      12. distribute-rgt-neg-in 62.2%

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

      13. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

      1. associate-/l/ 62.2%

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

      2. add-sqr-sqrt 33.0%

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

      3. sqrt-unprod 48.4%

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

      4. *-commutative 48.4%

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

      5. *-commutative 48.4%

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

      6. swap-sqr 48.4%

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

      7. metadata-eval 48.4%

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

      8. metadata-eval 48.4%

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

      9. swap-sqr 48.4%

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

      10. sqrt-unprod 25.6%

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

      11. add-sqr-sqrt 62.2%

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

      12. unpow2 62.2%

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

      13. associate-*r* 65.6%

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

      14. *-commutative 65.6%

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

      15. associate-/r* 65.6%

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

      16. *-un-lft-identity 65.6%

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

   6. Applied egg-rr 89.3%

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

      1. associate-*l/ 89.4%

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

      2. *-lft-identity 89.4%

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

      3. *-commutative 89.4%

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

      4. unpow2 89.4%

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

      5. rem-sqrt-square 89.4%

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

      6. *-commutative 89.4%

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

      7. unpow2 89.4%

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

      8. rem-sqrt-square 98.6%

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

      9. *-commutative 98.6%

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

   8. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. distribute-frac-neg2 98.6%

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

      3. *-commutative 98.6%

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

      4. distribute-neg-frac2 98.6%

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

      5. *-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. add-sqr-sqrt 59.6%

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

      8. fabs-sqr 59.6%

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

      9. add-sqr-sqrt 73.7%

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

      10. distribute-rgt-neg-in 73.7%

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

      11. add-sqr-sqrt 58.6%

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

      12. fabs-sqr 58.6%

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

      13. add-sqr-sqrt 98.6%

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

   10. Applied egg-rr 98.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 6.2 \cdot 10^{-236}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot \frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

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 (* c_m (* x_m s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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(c_m * Float64(x_m * s_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 = c_m * (x_m * s_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[(c$95$m * N[(x$95$m * s$95$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 68.5%

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

   1. associate-/l/ 68.5%

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

   2. remove-double-neg 68.5%

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

   3. distribute-frac-neg 68.5%

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

   4. distribute-neg-frac 68.5%

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

   5. remove-double-neg 68.5%

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

   6. *-commutative 68.5%

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

   7. associate-*r* 62.4%

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

   8. unpow2 62.4%

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

   9. associate-/r* 62.1%

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

   10. cos-neg 62.1%

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

   11. *-commutative 62.1%

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

   12. distribute-rgt-neg-in 62.1%

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

   13. metadata-eval 62.1%

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

3. Simplified 62.1%

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

   1. associate-/l/ 62.4%

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

   2. add-sqr-sqrt 31.5%

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

   3. sqrt-unprod 50.5%

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

   4. *-commutative 50.5%

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

   5. *-commutative 50.5%

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

   6. swap-sqr 50.5%

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

   7. metadata-eval 50.5%

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

   8. metadata-eval 50.5%

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

   9. swap-sqr 50.5%

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

   10. sqrt-unprod 27.4%

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

   11. add-sqr-sqrt 62.4%

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

   12. unpow2 62.4%

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

   13. associate-*r* 68.5%

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

   14. *-commutative 68.5%

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

   15. associate-/r* 68.5%

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

   16. *-un-lft-identity 68.5%

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

6. Applied egg-rr 87.2%

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

   1. associate-*l/ 87.3%

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

   2. *-lft-identity 87.3%

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

   3. *-commutative 87.3%

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

   4. unpow2 87.3%

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

   5. rem-sqrt-square 87.3%

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

   6. *-commutative 87.3%

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

   7. unpow2 87.3%

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

   8. rem-sqrt-square 97.5%

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

   9. *-commutative 97.5%

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

8. Simplified 97.5%

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

   1. frac-2neg 97.5%

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

   2. distribute-frac-neg2 97.5%

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

   3. *-commutative 97.5%

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

   4. distribute-neg-frac2 97.5%

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

   5. *-commutative 97.5%

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

   6. *-commutative 97.5%

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

   7. add-sqr-sqrt 56.2%

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

   8. fabs-sqr 56.2%

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

   9. add-sqr-sqrt 67.8%

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

   10. distribute-rgt-neg-in 67.8%

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

   11. add-sqr-sqrt 52.2%

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

   12. fabs-sqr 52.2%

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

   13. add-sqr-sqrt 97.5%

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

10. Applied egg-rr 97.5%

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

11. Final simplification 97.5%

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

Alternative 6: 80.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.6e+77)
   (/ (/ (/ 1.0 c_m) (* x_m s_m)) (* c_m (* x_m s_m)))
   (- (pow (* s_m (* x_m c_m)) -2.0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e+77) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = -pow((s_m * (x_m * c_m)), -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 3.6d+77) then
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
    else
        tmp = -((s_m * (x_m * c_m)) ** (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e+77) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = -Math.pow((s_m * (x_m * c_m)), -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 3.6e+77:
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
	else:
		tmp = -math.pow((s_m * (x_m * c_m)), -2.0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.6e+77)
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) / Float64(c_m * Float64(x_m * s_m)));
	else
		tmp = Float64(-(Float64(s_m * Float64(x_m * c_m)) ^ -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 3.6e+77)
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	else
		tmp = -((s_m * (x_m * c_m)) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.6e+77], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999998e77

   1. Initial program 68.6%

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

      1. associate-/l/ 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg 68.6%

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

      4. distribute-neg-frac 68.6%

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

      5. remove-double-neg 68.6%

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

      6. *-commutative 68.6%

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

      7. associate-*r* 63.9%

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

      8. unpow2 63.9%

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

      9. associate-/r* 63.6%

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

      10. cos-neg 63.6%

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

      11. *-commutative 63.6%

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

      12. distribute-rgt-neg-in 63.6%

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

      13. metadata-eval 63.6%

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

   3. Simplified 63.6%

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

      1. associate-/l/ 63.9%

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

      2. add-sqr-sqrt 39.2%

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

      3. sqrt-unprod 57.4%

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

      4. *-commutative 57.4%

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

      5. *-commutative 57.4%

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

      6. swap-sqr 57.4%

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

      7. metadata-eval 57.4%

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

      8. metadata-eval 57.4%

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

      9. swap-sqr 57.4%

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

      10. sqrt-unprod 21.4%

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

      11. add-sqr-sqrt 63.9%

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

      12. unpow2 63.9%

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

      13. associate-*r* 68.6%

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

      14. *-commutative 68.6%

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

      15. associate-/r* 68.6%

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

      16. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*l/ 86.1%

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

      2. *-lft-identity 86.1%

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

      3. *-commutative 86.1%

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

      4. unpow2 86.1%

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

      5. rem-sqrt-square 86.1%

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

      6. *-commutative 86.1%

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

      7. unpow2 86.1%

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

      8. rem-sqrt-square 97.0%

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

      9. *-commutative 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around 0 81.2%

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

       1. associate-/r* 81.3%

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

       2. *-commutative 81.3%

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

       3. rem-square-sqrt 48.2%

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

       4. fabs-sqr 48.2%

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

       5. rem-square-sqrt 56.7%

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

       6. *-commutative 56.7%

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

   11. Simplified 56.7%

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

       1. add-sqr-sqrt 43.3%

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

       2. fabs-sqr 43.3%

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

       3. add-sqr-sqrt 81.3%

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

   13. Applied egg-rr 81.3%

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

   if 3.5999999999999998e77 < x

   1. Initial program 68.3%

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

      1. associate-/l/ 68.3%

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

      2. remove-double-neg 68.3%

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

      3. distribute-frac-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. remove-double-neg 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-*r* 56.3%

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

      8. unpow2 56.3%

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

      9. associate-/r* 56.3%

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

      10. cos-neg 56.3%

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

      11. *-commutative 56.3%

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

      12. distribute-rgt-neg-in 56.3%

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

      13. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in x around 0 46.9%

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

      1. associate-/r* 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. unpow2 42.9%

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

      5. unpow2 42.9%

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

      6. swap-sqr 51.8%

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

      7. unpow2 51.8%

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

      8. associate-/r* 55.8%

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

      9. unpow2 55.8%

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

      10. rem-square-sqrt 55.8%

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

      11. swap-sqr 66.1%

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

      12. unpow2 66.1%

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

      13. unpow2 66.1%

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

      14. rem-sqrt-square 66.7%

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

      15. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. unpow-prod-down 55.8%

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

      2. pow2 55.8%

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

      3. pow2 55.8%

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

      4. sqr-abs 55.8%

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

      5. swap-sqr 66.7%

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

   9. Applied egg-rr 66.7%

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

   11. Applied egg-rr 72.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.6e+77)
   (/ (/ (/ 1.0 c_m) (* x_m s_m)) (* c_m (* x_m s_m)))
   (- (pow (* x_m (* c_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) {
	double tmp;
	if (x_m <= 3.6e+77) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = -pow((x_m * (c_m * s_m)), -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 3.6d+77) then
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
    else
        tmp = -((x_m * (c_m * s_m)) ** (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.6e+77) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = -Math.pow((x_m * (c_m * s_m)), -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 3.6e+77:
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m))
	else:
		tmp = -math.pow((x_m * (c_m * s_m)), -2.0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.6e+77)
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) / Float64(c_m * Float64(x_m * s_m)));
	else
		tmp = Float64(-(Float64(x_m * Float64(c_m * s_m)) ^ -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 3.6e+77)
		tmp = ((1.0 / c_m) / (x_m * s_m)) / (c_m * (x_m * s_m));
	else
		tmp = -((x_m * (c_m * s_m)) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.6e+77], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999998e77

   1. Initial program 68.6%

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

      1. associate-/l/ 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg 68.6%

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

      4. distribute-neg-frac 68.6%

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

      5. remove-double-neg 68.6%

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

      6. *-commutative 68.6%

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

      7. associate-*r* 63.9%

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

      8. unpow2 63.9%

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

      9. associate-/r* 63.6%

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

      10. cos-neg 63.6%

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

      11. *-commutative 63.6%

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

      12. distribute-rgt-neg-in 63.6%

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

      13. metadata-eval 63.6%

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

   3. Simplified 63.6%

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

      1. associate-/l/ 63.9%

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

      2. add-sqr-sqrt 39.2%

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

      3. sqrt-unprod 57.4%

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

      4. *-commutative 57.4%

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

      5. *-commutative 57.4%

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

      6. swap-sqr 57.4%

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

      7. metadata-eval 57.4%

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

      8. metadata-eval 57.4%

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

      9. swap-sqr 57.4%

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

      10. sqrt-unprod 21.4%

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

      11. add-sqr-sqrt 63.9%

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

      12. unpow2 63.9%

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

      13. associate-*r* 68.6%

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

      14. *-commutative 68.6%

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

      15. associate-/r* 68.6%

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

      16. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*l/ 86.1%

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

      2. *-lft-identity 86.1%

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

      3. *-commutative 86.1%

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

      4. unpow2 86.1%

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

      5. rem-sqrt-square 86.1%

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

      6. *-commutative 86.1%

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

      7. unpow2 86.1%

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

      8. rem-sqrt-square 97.0%

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

      9. *-commutative 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around 0 81.2%

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

       1. associate-/r* 81.3%

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

       2. *-commutative 81.3%

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

       3. rem-square-sqrt 48.2%

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

       4. fabs-sqr 48.2%

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

       5. rem-square-sqrt 56.7%

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

       6. *-commutative 56.7%

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

   11. Simplified 56.7%

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

       1. add-sqr-sqrt 43.3%

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

       2. fabs-sqr 43.3%

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

       3. add-sqr-sqrt 81.3%

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

   13. Applied egg-rr 81.3%

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

   if 3.5999999999999998e77 < x

   1. Initial program 68.3%

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

      1. associate-/l/ 68.3%

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

      2. remove-double-neg 68.3%

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

      3. distribute-frac-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. remove-double-neg 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-*r* 56.3%

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

      8. unpow2 56.3%

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

      9. associate-/r* 56.3%

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

      10. cos-neg 56.3%

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

      11. *-commutative 56.3%

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

      12. distribute-rgt-neg-in 56.3%

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

      13. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in x around 0 46.9%

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

      1. associate-/r* 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. unpow2 42.9%

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

      5. unpow2 42.9%

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

      6. swap-sqr 51.8%

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

      7. unpow2 51.8%

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

      8. associate-/r* 55.8%

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

      9. unpow2 55.8%

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

      10. rem-square-sqrt 55.8%

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

      11. swap-sqr 66.1%

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

      12. unpow2 66.1%

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

      13. unpow2 66.1%

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

      14. rem-sqrt-square 66.7%

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

      15. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. unpow-prod-down 55.8%

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

      2. pow2 55.8%

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

      3. pow2 55.8%

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

      4. sqr-abs 55.8%

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

      5. swap-sqr 66.7%

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

   9. Applied egg-rr 66.7%

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

       1. inv-pow 66.7%

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

       2. swap-sqr 55.8%

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

       3. unpow-prod-down 51.8%

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

       4. *-commutative 51.8%

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

       5. associate-*r* 51.4%

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

       6. associate-*r* 46.9%

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

       7. unpow2 46.9%

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

       8. unpow-prod-down 50.9%

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

       9. inv-pow 50.9%

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

       10. frac-2neg 50.9%

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

       11. metadata-eval 50.9%

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

       12. div-inv 50.9%

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

       13. associate-*l* 61.2%

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

       14. distribute-lft-neg-in 61.2%

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

   11. Applied egg-rr 72.4%

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

       1. neg-mul-1 72.4%

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

       2. *-commutative 72.4%

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

       3. associate-*l* 72.2%

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

   13. Simplified 72.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

   1. associate-/l/ 68.5%

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

   2. remove-double-neg 68.5%

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

   3. distribute-frac-neg 68.5%

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

   4. distribute-neg-frac 68.5%

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

   5. remove-double-neg 68.5%

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

   6. *-commutative 68.5%

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

   7. associate-*r* 62.4%

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

   8. unpow2 62.4%

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

   9. associate-/r* 62.1%

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

   10. cos-neg 62.1%

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

   11. *-commutative 62.1%

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

   12. distribute-rgt-neg-in 62.1%

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

   13. metadata-eval 62.1%

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

3. Simplified 62.1%

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

   1. associate-/l/ 62.4%

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

   2. add-sqr-sqrt 31.5%

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

   3. sqrt-unprod 50.5%

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

   4. *-commutative 50.5%

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

   5. *-commutative 50.5%

      \[\leadsto \frac{\frac{\cos \left(\sqrt{\left(-2 \cdot x\right) \cdot \color{blue}{\left(-2 \cdot x\right)}}\right)}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   6. swap-sqr 50.5%

      \[\leadsto \frac{\frac{\cos \left(\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(x \cdot x\right)}}\right)}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   7. metadata-eval 50.5%

      \[\leadsto \frac{\frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   8. metadata-eval 50.5%

      \[\leadsto \frac{\frac{\cos \left(\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(x \cdot x\right)}\right)}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   9. swap-sqr 50.5%

      \[\leadsto \frac{\frac{\cos \left(\sqrt{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}\right)}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   10. sqrt-unprod 27.4%

       \[\leadsto \frac{\frac{\cos \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)}}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   11. add-sqr-sqrt 62.4%

       \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   12. unpow2 62.4%

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}}}{{c}^{2}} \]

   13. associate-*r* 68.5%

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({s}^{2} \cdot x\right) \cdot x}}}{{c}^{2}} \]

   14. *-commutative 68.5%

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x}}{{c}^{2}} \]

   15. associate-/r* 68.5%

       \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   16. *-un-lft-identity 68.5%

       \[\leadsto \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}} \]

6. Applied egg-rr 87.2%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}} \]
7. Step-by-step derivation

   1. associate-*l/ 87.3%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}} \]

   2. *-lft-identity 87.3%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \]

   3. *-commutative 87.3%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \]

   4. unpow2 87.3%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \sqrt{\color{blue}{\left(s \cdot x\right) \cdot \left(s \cdot x\right)}}}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \]

   5. rem-sqrt-square 87.3%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \color{blue}{\left|s \cdot x\right|}}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \]

   6. *-commutative 87.3%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left|\color{blue}{x \cdot s}\right|}}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \]

   7. unpow2 87.3%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left|x \cdot s\right|}}{c \cdot \sqrt{\color{blue}{\left(s \cdot x\right) \cdot \left(s \cdot x\right)}}} \]

   8. rem-sqrt-square 97.5%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left|x \cdot s\right|}}{c \cdot \color{blue}{\left|s \cdot x\right|}} \]

   9. *-commutative 97.5%

      \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left|x \cdot s\right|}}{c \cdot \left|\color{blue}{x \cdot s}\right|} \]

8. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left|x \cdot s\right|}}{c \cdot \left|x \cdot s\right|}} \]

9. Taylor expanded in x around 0 78.4%

   \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left|s \cdot x\right|}}}{c \cdot \left|x \cdot s\right|} \]
10. Step-by-step derivation

    1. associate-/r* 78.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{c}}{\left|s \cdot x\right|}}}{c \cdot \left|x \cdot s\right|} \]

    2. *-commutative 78.4%

       \[\leadsto \frac{\frac{\frac{1}{c}}{\left|\color{blue}{x \cdot s}\right|}}{c \cdot \left|x \cdot s\right|} \]

    3. rem-square-sqrt 44.6%

       \[\leadsto \frac{\frac{\frac{1}{c}}{\left|\color{blue}{\sqrt{x \cdot s} \cdot \sqrt{x \cdot s}}\right|}}{c \cdot \left|x \cdot s\right|} \]

    4. fabs-sqr 44.6%

       \[\leadsto \frac{\frac{\frac{1}{c}}{\color{blue}{\sqrt{x \cdot s} \cdot \sqrt{x \cdot s}}}}{c \cdot \left|x \cdot s\right|} \]

    5. rem-square-sqrt 58.3%

       \[\leadsto \frac{\frac{\frac{1}{c}}{\color{blue}{x \cdot s}}}{c \cdot \left|x \cdot s\right|} \]

    6. *-commutative 58.3%

       \[\leadsto \frac{\frac{\frac{1}{c}}{\color{blue}{s \cdot x}}}{c \cdot \left|x \cdot s\right|} \]

11. Simplified 58.3%

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{c}}{s \cdot x}}}{c \cdot \left|x \cdot s\right|} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 40.7%

       \[\leadsto \frac{\frac{\frac{1}{c}}{s \cdot x}}{c \cdot \left|\color{blue}{\sqrt{x \cdot s} \cdot \sqrt{x \cdot s}}\right|} \]

    2. fabs-sqr 40.7%

       \[\leadsto \frac{\frac{\frac{1}{c}}{s \cdot x}}{c \cdot \color{blue}{\left(\sqrt{x \cdot s} \cdot \sqrt{x \cdot s}\right)}} \]

    3. add-sqr-sqrt 78.4%

       \[\leadsto \frac{\frac{\frac{1}{c}}{s \cdot x}}{c \cdot \color{blue}{\left(x \cdot s\right)}} \]

13. Applied egg-rr 78.4%

    \[\leadsto \frac{\frac{\frac{1}{c}}{s \cdot x}}{c \cdot \color{blue}{\left(x \cdot s\right)}} \]

14. Final simplification 78.4%

    \[\leadsto \frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)} \]
15. Add Preprocessing

Alternative 9: 80.1% 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 68.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 68.5%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot x}}{{c}^{2}}} \]

   2. remove-double-neg 68.5%

      \[\leadsto \frac{\color{blue}{-\left(-\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot x}\right)}}{{c}^{2}} \]

   3. distribute-frac-neg 68.5%

      \[\leadsto \frac{-\color{blue}{\frac{-\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot x}}}{{c}^{2}} \]

   4. distribute-neg-frac 68.5%

      \[\leadsto \frac{\color{blue}{\frac{-\left(-\cos \left(2 \cdot x\right)\right)}{\left(x \cdot {s}^{2}\right) \cdot x}}}{{c}^{2}} \]

   5. remove-double-neg 68.5%

      \[\leadsto \frac{\frac{\color{blue}{\cos \left(2 \cdot x\right)}}{\left(x \cdot {s}^{2}\right) \cdot x}}{{c}^{2}} \]

   6. *-commutative 68.5%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(x \cdot {s}^{2}\right)}}}{{c}^{2}} \]

   7. associate-*r* 62.4%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot {s}^{2}}}}{{c}^{2}} \]

   8. unpow2 62.4%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{{x}^{2}} \cdot {s}^{2}}}{{c}^{2}} \]

   9. associate-/r* 62.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}} \]

   10. cos-neg 62.1%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{x}^{2}}}{{s}^{2}}}{{c}^{2}} \]

   11. *-commutative 62.1%

       \[\leadsto \frac{\frac{\frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{x}^{2}}}{{s}^{2}}}{{c}^{2}} \]

   12. distribute-rgt-neg-in 62.1%

       \[\leadsto \frac{\frac{\frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{x}^{2}}}{{s}^{2}}}{{c}^{2}} \]

   13. metadata-eval 62.1%

       \[\leadsto \frac{\frac{\frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{x}^{2}}}{{s}^{2}}}{{c}^{2}} \]

3. Simplified 62.1%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}{{c}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 54.7%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 53.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 53.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. *-commutative 53.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot {x}^{2}}} \]

   4. unpow2 53.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}} \]

   5. unpow2 53.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   6. swap-sqr 64.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{\left(s \cdot x\right) \cdot \left(s \cdot x\right)}} \]

   7. unpow2 64.7%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{\left(s \cdot x\right)}^{2}}} \]

   8. associate-/r* 65.6%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot {\left(s \cdot x\right)}^{2}}} \]

   9. unpow2 65.6%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(s \cdot x\right)}^{2}} \]

   10. rem-square-sqrt 65.6%

       \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\sqrt{{\left(s \cdot x\right)}^{2}} \cdot \sqrt{{\left(s \cdot x\right)}^{2}}\right)}} \]

   11. swap-sqr 72.1%

       \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}\right) \cdot \left(c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}\right)}} \]

   12. unpow2 72.1%

       \[\leadsto \frac{1}{\color{blue}{{\left(c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}\right)}^{2}}} \]

   13. unpow2 72.1%

       \[\leadsto \frac{1}{{\left(c \cdot \sqrt{\color{blue}{\left(s \cdot x\right) \cdot \left(s \cdot x\right)}}\right)}^{2}} \]

   14. rem-sqrt-square 78.4%

       \[\leadsto \frac{1}{{\left(c \cdot \color{blue}{\left|s \cdot x\right|}\right)}^{2}} \]

   15. *-commutative 78.4%

       \[\leadsto \frac{1}{{\left(c \cdot \left|\color{blue}{x \cdot s}\right|\right)}^{2}} \]

7. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left|x \cdot s\right|\right)}^{2}}} \]
8. Step-by-step derivation

   1. unpow-prod-down 65.6%

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot {\left(\left|x \cdot s\right|\right)}^{2}}} \]

   2. pow2 65.6%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot {\left(\left|x \cdot s\right|\right)}^{2}} \]

   3. pow2 65.6%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left|x \cdot s\right| \cdot \left|x \cdot s\right|\right)}} \]

   4. sqr-abs 65.6%

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   5. swap-sqr 78.4%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

9. Applied egg-rr 78.4%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))