mixedcos

Percentage Accurate: 66.4% → 98.1%

Time: 13.9s

Alternatives: 11

Speedup: 24.1×

• 32.5% 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: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot c\_m} \cdot \frac{\frac{1}{s\_m} \cdot \frac{\frac{1}{c\_m}}{x\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.3e-6)
   (/ (cos (* x_m -2.0)) (pow (* c_m (* x_m s_m)) 2.0))
   (*
    (/ (cos (* x_m 2.0)) (* x_m c_m))
    (/ (* (/ 1.0 s_m) (/ (/ 1.0 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) {
	double tmp;
	if (x_m <= 1.3e-6) {
		tmp = cos((x_m * -2.0)) / pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = (cos((x_m * 2.0)) / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.3d-6) then
        tmp = cos((x_m * (-2.0d0))) / ((c_m * (x_m * s_m)) ** 2.0d0)
    else
        tmp = (cos((x_m * 2.0d0)) / (x_m * c_m)) * (((1.0d0 / s_m) * ((1.0d0 / c_m) / x_m)) / s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.3e-6) {
		tmp = Math.cos((x_m * -2.0)) / Math.pow((c_m * (x_m * s_m)), 2.0);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.3e-6:
		tmp = math.cos((x_m * -2.0)) / math.pow((c_m * (x_m * s_m)), 2.0)
	else:
		tmp = (math.cos((x_m * 2.0)) / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.3e-6)
		tmp = Float64(cos(Float64(x_m * -2.0)) / (Float64(c_m * Float64(x_m * s_m)) ^ 2.0));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * c_m)) * Float64(Float64(Float64(1.0 / s_m) * Float64(Float64(1.0 / c_m) / x_m)) / s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.3e-6)
		tmp = cos((x_m * -2.0)) / ((c_m * (x_m * s_m)) ^ 2.0);
	else
		tmp = (cos((x_m * 2.0)) / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.3e-6], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / s$95$m), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000005e-6

   1. Initial program 69.1%

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

      1. associate-/r* 68.6%

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

      2. cos-neg 68.6%

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

      3. distribute-rgt-neg-out 68.6%

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

      4. distribute-rgt-neg-out 68.6%

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

      5. *-commutative 68.6%

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

      6. distribute-rgt-neg-in 68.6%

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

      7. metadata-eval 68.6%

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

      8. *-commutative 68.6%

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

      9. associate-*l* 64.3%

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

      10. unpow2 64.3%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in x around inf 64.8%

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

      1. associate-/r* 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

      4. unpow2 64.3%

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

      5. swap-sqr 78.4%

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

      6. unpow2 78.4%

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

      7. associate-/r* 78.9%

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

      8. *-commutative 78.9%

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

      9. unpow2 78.9%

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

      10. unpow2 78.9%

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

      11. swap-sqr 97.8%

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

      12. unpow2 97.8%

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

      13. *-commutative 97.8%

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

   7. Simplified 97.8%

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

   if 1.30000000000000005e-6 < x

   1. Initial program 75.1%

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

      1. associate-/l/ 75.2%

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

      2. remove-double-neg 75.2%

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

      3. distribute-frac-neg 75.2%

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

      4. distribute-neg-frac 75.2%

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

      5. remove-double-neg 75.2%

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

      6. *-commutative 75.2%

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

      7. associate-*r* 69.2%

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

      8. unpow2 69.2%

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

      9. associate-/r* 69.2%

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

      10. cos-neg 69.2%

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

      11. *-commutative 69.2%

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

      12. distribute-rgt-neg-in 69.2%

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

      13. metadata-eval 69.2%

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

   3. Simplified 69.2%

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

      1. associate-/l/ 69.2%

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

      2. *-un-lft-identity 69.2%

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

      3. add-sqr-sqrt 69.1%

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

      4. times-frac 69.1%

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

      5. pow-prod-down 69.2%

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

      6. sqrt-pow1 55.6%

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

      7. metadata-eval 55.6%

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

      8. pow1 55.6%

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

      9. *-commutative 55.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 26.0%

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

      12. *-commutative 26.0%

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

      13. *-commutative 26.0%

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

      14. swap-sqr 26.0%

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

      15. metadata-eval 26.0%

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

      16. metadata-eval 26.0%

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

      17. swap-sqr 26.0%

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

      18. sqrt-unprod 56.0%

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

      19. add-sqr-sqrt 55.6%

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

   6. Applied egg-rr 82.7%

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

      1. div-inv 82.7%

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

      2. frac-times 82.6%

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

      3. *-un-lft-identity 82.6%

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

      4. pow2 82.6%

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

      5. frac-times 82.6%

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

      6. unpow-prod-down 95.9%

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

      7. *-commutative 95.9%

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

      8. unpow2 95.9%

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

      9. frac-times 96.0%

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

      10. div-inv 95.9%

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

      11. div-inv 95.9%

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

      12. associate-*r* 94.2%

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

      13. times-frac 88.7%

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

      14. *-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. *-commutative 88.7%

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

      3. times-frac 92.2%

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

   10. Applied egg-rr 92.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot c} \cdot \frac{\frac{1}{s} \cdot \frac{\frac{1}{c}}{x}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot c\_m} \cdot \frac{\frac{1}{s\_m} \cdot \frac{\frac{1}{c\_m}}{x\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 1.35e-6)
     (/ (/ t_0 t_1) t_1)
     (* (/ t_0 (* x_m c_m)) (/ (* (/ 1.0 s_m) (/ (/ 1.0 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) {
	double t_0 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.35e-6) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = (t_0 / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 1.35d-6) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = (t_0 / (x_m * c_m)) * (((1.0d0 / s_m) * ((1.0d0 / c_m) / x_m)) / s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.35e-6) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = (t_0 / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.35e-6:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = (t_0 / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.35e-6)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(t_0 / Float64(x_m * c_m)) * Float64(Float64(Float64(1.0 / s_m) * Float64(Float64(1.0 / c_m) / x_m)) / s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.35e-6)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = (t_0 / (x_m * c_m)) * (((1.0 / s_m) * ((1.0 / c_m) / x_m)) / s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.35e-6], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t$95$0 / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / s$95$m), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999999e-6

   1. Initial program 69.1%

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

      1. *-un-lft-identity 69.1%

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

      2. add-sqr-sqrt 69.0%

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

      3. times-frac 69.0%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. div-inv 97.8%

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

      3. div-inv 97.8%

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

      4. div-inv 97.8%

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

      5. *-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 1.34999999999999999e-6 < x

   1. Initial program 75.1%

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

      1. associate-/l/ 75.2%

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

      2. remove-double-neg 75.2%

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

      3. distribute-frac-neg 75.2%

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

      4. distribute-neg-frac 75.2%

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

      5. remove-double-neg 75.2%

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

      6. *-commutative 75.2%

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

      7. associate-*r* 69.2%

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

      8. unpow2 69.2%

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

      9. associate-/r* 69.2%

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

      10. cos-neg 69.2%

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

      11. *-commutative 69.2%

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

      12. distribute-rgt-neg-in 69.2%

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

      13. metadata-eval 69.2%

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

   3. Simplified 69.2%

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

      1. associate-/l/ 69.2%

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

      2. *-un-lft-identity 69.2%

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

      3. add-sqr-sqrt 69.1%

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

      4. times-frac 69.1%

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

      5. pow-prod-down 69.2%

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

      6. sqrt-pow1 55.6%

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

      7. metadata-eval 55.6%

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

      8. pow1 55.6%

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

      9. *-commutative 55.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 26.0%

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

      12. *-commutative 26.0%

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

      13. *-commutative 26.0%

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

      14. swap-sqr 26.0%

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

      15. metadata-eval 26.0%

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

      16. metadata-eval 26.0%

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

      17. swap-sqr 26.0%

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

      18. sqrt-unprod 56.0%

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

      19. add-sqr-sqrt 55.6%

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

   6. Applied egg-rr 82.7%

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

      1. div-inv 82.7%

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

      2. frac-times 82.6%

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

      3. *-un-lft-identity 82.6%

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

      4. pow2 82.6%

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

      5. frac-times 82.6%

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

      6. unpow-prod-down 95.9%

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

      7. *-commutative 95.9%

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

      8. unpow2 95.9%

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

      9. frac-times 96.0%

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

      10. div-inv 95.9%

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

      11. div-inv 95.9%

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

      12. associate-*r* 94.2%

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

      13. times-frac 88.7%

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

      14. *-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. *-commutative 88.7%

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

      3. times-frac 92.2%

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

   10. Applied egg-rr 92.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot c} \cdot \frac{\frac{1}{s} \cdot \frac{\frac{1}{c}}{x}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{c\_m}{\cos \left(x\_m \cdot 2\right)}\\ \mathbf{if}\;c\_m \leq 2.15 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(c\_m \cdot s\_m\right)}}{x\_m \cdot \left(s\_m \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}}{\left(x\_m \cdot s\_m\right) \cdot t\_0}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0)))))
   (if (<= c_m 2.15e-280)
     (/ (/ 1.0 (* x_m (* c_m s_m))) (* x_m (* s_m t_0)))
     (/ (/ (/ 1.0 c_m) (* x_m s_m)) (* (* x_m s_m) t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m / cos((x_m * 2.0));
	double tmp;
	if (c_m <= 2.15e-280) {
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * t_0));
	} else {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / ((x_m * s_m) * t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m / cos((x_m * 2.0d0))
    if (c_m <= 2.15d-280) then
        tmp = (1.0d0 / (x_m * (c_m * s_m))) / (x_m * (s_m * t_0))
    else
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) / ((x_m * s_m) * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m / Math.cos((x_m * 2.0));
	double tmp;
	if (c_m <= 2.15e-280) {
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * t_0));
	} else {
		tmp = ((1.0 / c_m) / (x_m * s_m)) / ((x_m * s_m) * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m / math.cos((x_m * 2.0))
	tmp = 0
	if c_m <= 2.15e-280:
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * t_0))
	else:
		tmp = ((1.0 / c_m) / (x_m * s_m)) / ((x_m * s_m) * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m / cos(Float64(x_m * 2.0)))
	tmp = 0.0
	if (c_m <= 2.15e-280)
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(c_m * s_m))) / Float64(x_m * Float64(s_m * t_0)));
	else
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) / Float64(Float64(x_m * s_m) * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m / cos((x_m * 2.0));
	tmp = 0.0;
	if (c_m <= 2.15e-280)
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * t_0));
	else
		tmp = ((1.0 / c_m) / (x_m * s_m)) / ((x_m * s_m) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m / N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 2.15e-280], N[(N[(1.0 / N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(s$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.1499999999999999e-280

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. remove-double-neg 69.8%

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

      3. distribute-frac-neg 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. remove-double-neg 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-*r* 67.3%

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

      8. unpow2 67.3%

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

      9. associate-/r* 66.5%

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

      10. cos-neg 66.5%

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

      11. *-commutative 66.5%

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

      12. distribute-rgt-neg-in 66.5%

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

      13. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

      1. associate-/l/ 67.3%

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

      2. *-un-lft-identity 67.3%

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

      3. add-sqr-sqrt 67.2%

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

      4. times-frac 67.3%

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

      5. pow-prod-down 67.3%

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

      6. sqrt-pow1 47.8%

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

      7. metadata-eval 47.8%

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

      8. pow1 47.8%

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

      9. *-commutative 47.8%

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

      10. add-sqr-sqrt 22.2%

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

      11. sqrt-unprod 33.7%

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

      12. *-commutative 33.7%

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

      13. *-commutative 33.7%

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

      14. swap-sqr 33.7%

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

      15. metadata-eval 33.7%

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

      16. metadata-eval 33.7%

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

      17. swap-sqr 33.7%

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

      18. sqrt-unprod 26.7%

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

      19. add-sqr-sqrt 47.8%

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

   6. Applied egg-rr 79.7%

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

      1. div-inv 78.9%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. pow2 78.7%

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

      5. frac-times 79.5%

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

      6. unpow-prod-down 96.0%

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

      7. *-commutative 96.0%

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

      8. unpow2 96.0%

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

      9. frac-times 96.0%

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

      10. div-inv 96.0%

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

      11. div-inv 96.0%

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

      12. *-commutative 96.0%

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

      13. associate-*r* 91.7%

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

      14. times-frac 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l/ 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-/l/ 84.8%

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

      4. associate-/l/ 84.8%

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

   10. Simplified 84.8%

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

       1. clear-num 84.8%

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

       2. frac-times 91.7%

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

       3. div-inv 91.7%

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

       4. *-un-lft-identity 91.7%

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

       5. associate-/l/ 91.7%

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

       6. frac-times 91.7%

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

       7. metadata-eval 91.7%

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

       8. *-commutative 91.7%

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

       9. *-commutative 91.7%

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

       10. associate-*r* 95.2%

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

       11. *-commutative 95.2%

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

       12. *-un-lft-identity 95.2%

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

       13. times-frac 95.3%

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

       14. /-rgt-identity 95.3%

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

   12. Applied egg-rr 95.3%

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

   if 2.1499999999999999e-280 < c

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. times-frac 70.7%

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

   4. Applied egg-rr 98.9%

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

      1. clear-num 98.9%

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

      2. un-div-inv 98.9%

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

      3. associate-/r* 98.9%

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

      4. *-commutative 98.9%

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

      5. *-un-lft-identity 98.9%

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

      6. times-frac 98.9%

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

      7. /-rgt-identity 98.9%

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

      8. *-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.15 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(s \cdot \frac{c}{\cos \left(x \cdot 2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{\left(x \cdot s\right) \cdot \frac{c}{\cos \left(x \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 1.45 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(c\_m \cdot s\_m\right)}}{x\_m \cdot \left(s\_m \cdot \frac{c\_m}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1} \cdot \frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= c_m 1.45e-280)
     (/ (/ 1.0 (* x_m (* c_m s_m))) (* x_m (* s_m (/ c_m t_0))))
     (* (/ 1.0 t_1) (/ t_0 t_1)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (c_m <= 1.45e-280) {
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * (c_m / t_0)));
	} else {
		tmp = (1.0 / t_1) * (t_0 / t_1);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (c_m <= 1.45d-280) then
        tmp = (1.0d0 / (x_m * (c_m * s_m))) / (x_m * (s_m * (c_m / t_0)))
    else
        tmp = (1.0d0 / t_1) * (t_0 / t_1)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (c_m <= 1.45e-280) {
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * (c_m / t_0)));
	} else {
		tmp = (1.0 / t_1) * (t_0 / t_1);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if c_m <= 1.45e-280:
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * (c_m / t_0)))
	else:
		tmp = (1.0 / t_1) * (t_0 / t_1)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (c_m <= 1.45e-280)
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(c_m * s_m))) / Float64(x_m * Float64(s_m * Float64(c_m / t_0))));
	else
		tmp = Float64(Float64(1.0 / t_1) * Float64(t_0 / t_1));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (c_m <= 1.45e-280)
		tmp = (1.0 / (x_m * (c_m * s_m))) / (x_m * (s_m * (c_m / t_0)));
	else
		tmp = (1.0 / t_1) * (t_0 / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 1.45e-280], N[(N[(1.0 / N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(s$95$m * N[(c$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.45e-280

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. remove-double-neg 69.8%

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

      3. distribute-frac-neg 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. remove-double-neg 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-*r* 67.3%

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

      8. unpow2 67.3%

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

      9. associate-/r* 66.5%

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

      10. cos-neg 66.5%

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

      11. *-commutative 66.5%

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

      12. distribute-rgt-neg-in 66.5%

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

      13. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

      1. associate-/l/ 67.3%

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

      2. *-un-lft-identity 67.3%

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

      3. add-sqr-sqrt 67.2%

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

      4. times-frac 67.3%

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

      5. pow-prod-down 67.3%

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

      6. sqrt-pow1 47.8%

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

      7. metadata-eval 47.8%

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

      8. pow1 47.8%

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

      9. *-commutative 47.8%

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

      10. add-sqr-sqrt 22.2%

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

      11. sqrt-unprod 33.7%

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

      12. *-commutative 33.7%

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

      13. *-commutative 33.7%

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

      14. swap-sqr 33.7%

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

      15. metadata-eval 33.7%

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

      16. metadata-eval 33.7%

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

      17. swap-sqr 33.7%

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

      18. sqrt-unprod 26.7%

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

      19. add-sqr-sqrt 47.8%

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

   6. Applied egg-rr 79.7%

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

      1. div-inv 78.9%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. pow2 78.7%

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

      5. frac-times 79.5%

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

      6. unpow-prod-down 96.0%

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

      7. *-commutative 96.0%

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

      8. unpow2 96.0%

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

      9. frac-times 96.0%

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

      10. div-inv 96.0%

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

      11. div-inv 96.0%

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

      12. *-commutative 96.0%

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

      13. associate-*r* 91.7%

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

      14. times-frac 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l/ 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-/l/ 84.8%

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

      4. associate-/l/ 84.8%

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

   10. Simplified 84.8%

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

       1. clear-num 84.8%

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

       2. frac-times 91.7%

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

       3. div-inv 91.7%

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

       4. *-un-lft-identity 91.7%

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

       5. associate-/l/ 91.7%

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

       6. frac-times 91.7%

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

       7. metadata-eval 91.7%

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

       8. *-commutative 91.7%

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

       9. *-commutative 91.7%

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

       10. associate-*r* 95.2%

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

       11. *-commutative 95.2%

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

       12. *-un-lft-identity 95.2%

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

       13. times-frac 95.3%

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

       14. /-rgt-identity 95.3%

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

   12. Applied egg-rr 95.3%

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

   if 1.45e-280 < c

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. times-frac 70.7%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.45 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(s \cdot \frac{c}{\cos \left(x \cdot 2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 2.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m \cdot \left(c\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{t\_1}{t\_0}\\ \end{array} \end{array} \]

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 2.3e-280)
     (/ (/ (/ t_1 (* x_m (* c_m s_m))) (* c_m s_m)) x_m)
     (* (/ 1.0 t_0) (/ t_1 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 <= 2.3e-280) {
		tmp = ((t_1 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m;
	} else {
		tmp = (1.0 / t_0) * (t_1 / 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 <= 2.3d-280) then
        tmp = ((t_1 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m
    else
        tmp = (1.0d0 / t_0) * (t_1 / 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 <= 2.3e-280) {
		tmp = ((t_1 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m;
	} else {
		tmp = (1.0 / t_0) * (t_1 / 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 <= 2.3e-280:
		tmp = ((t_1 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m
	else:
		tmp = (1.0 / t_0) * (t_1 / 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 <= 2.3e-280)
		tmp = Float64(Float64(Float64(t_1 / Float64(x_m * Float64(c_m * s_m))) / Float64(c_m * s_m)) / x_m);
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / 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 <= 2.3e-280)
		tmp = ((t_1 / (x_m * (c_m * s_m))) / (c_m * s_m)) / x_m;
	else
		tmp = (1.0 / t_0) * (t_1 / 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, 2.3e-280], N[(N[(N[(t$95$1 / 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], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.3e-280

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. remove-double-neg 69.8%

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

      3. distribute-frac-neg 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. remove-double-neg 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-*r* 67.3%

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

      8. unpow2 67.3%

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

      9. associate-/r* 66.5%

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

      10. cos-neg 66.5%

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

      11. *-commutative 66.5%

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

      12. distribute-rgt-neg-in 66.5%

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

      13. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

      1. associate-/l/ 67.3%

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

      2. *-un-lft-identity 67.3%

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

      3. add-sqr-sqrt 67.2%

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

      4. times-frac 67.3%

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

      5. pow-prod-down 67.3%

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

      6. sqrt-pow1 47.8%

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

      7. metadata-eval 47.8%

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

      8. pow1 47.8%

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

      9. *-commutative 47.8%

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

      10. add-sqr-sqrt 22.2%

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

      11. sqrt-unprod 33.7%

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

      12. *-commutative 33.7%

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

      13. *-commutative 33.7%

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

      14. swap-sqr 33.7%

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

      15. metadata-eval 33.7%

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

      16. metadata-eval 33.7%

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

      17. swap-sqr 33.7%

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

      18. sqrt-unprod 26.7%

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

      19. add-sqr-sqrt 47.8%

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

   6. Applied egg-rr 79.7%

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

      1. div-inv 78.9%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. pow2 78.7%

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

      5. frac-times 79.5%

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

      6. unpow-prod-down 96.0%

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

      7. *-commutative 96.0%

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

      8. unpow2 96.0%

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

      9. frac-times 96.0%

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

      10. div-inv 96.0%

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

      11. div-inv 96.0%

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

      12. *-commutative 96.0%

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

      13. associate-*r* 91.7%

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

      14. times-frac 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l/ 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-/l/ 84.8%

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

      4. associate-/l/ 84.8%

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

   10. Simplified 84.8%

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

       1. associate-*r/ 88.8%

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

       2. associate-/r* 88.8%

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

       3. frac-times 88.8%

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

       4. associate-/l/ 88.8%

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

       5. div-inv 88.8%

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

       6. associate-/r* 88.8%

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

       7. *-commutative 88.8%

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

       8. associate-*r* 92.4%

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

       9. *-commutative 92.4%

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

   12. Applied egg-rr 92.4%

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

   if 2.3e-280 < c

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. times-frac 70.7%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 1.12 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m \cdot \left(c\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 1.12e-281)
     (/ (/ (/ t_1 (* 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 <= 1.12e-281) {
		tmp = ((t_1 / (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 <= 1.12d-281) then
        tmp = ((t_1 / (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 <= 1.12e-281) {
		tmp = ((t_1 / (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 <= 1.12e-281:
		tmp = ((t_1 / (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 <= 1.12e-281)
		tmp = Float64(Float64(Float64(t_1 / 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 <= 1.12e-281)
		tmp = ((t_1 / (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, 1.12e-281], N[(N[(N[(t$95$1 / 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], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.12e-281

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. remove-double-neg 69.8%

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

      3. distribute-frac-neg 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. remove-double-neg 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-*r* 67.3%

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

      8. unpow2 67.3%

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

      9. associate-/r* 66.5%

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

      10. cos-neg 66.5%

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

      11. *-commutative 66.5%

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

      12. distribute-rgt-neg-in 66.5%

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

      13. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

      1. associate-/l/ 67.3%

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

      2. *-un-lft-identity 67.3%

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

      3. add-sqr-sqrt 67.2%

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

      4. times-frac 67.3%

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

      5. pow-prod-down 67.3%

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

      6. sqrt-pow1 47.8%

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

      7. metadata-eval 47.8%

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

      8. pow1 47.8%

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

      9. *-commutative 47.8%

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

      10. add-sqr-sqrt 22.2%

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

      11. sqrt-unprod 33.7%

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

      12. *-commutative 33.7%

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

      13. *-commutative 33.7%

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

      14. swap-sqr 33.7%

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

      15. metadata-eval 33.7%

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

      16. metadata-eval 33.7%

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

      17. swap-sqr 33.7%

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

      18. sqrt-unprod 26.7%

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

      19. add-sqr-sqrt 47.8%

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

   6. Applied egg-rr 79.7%

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

      1. div-inv 78.9%

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

      2. frac-times 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. pow2 78.7%

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

      5. frac-times 79.5%

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

      6. unpow-prod-down 96.0%

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

      7. *-commutative 96.0%

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

      8. unpow2 96.0%

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

      9. frac-times 96.0%

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

      10. div-inv 96.0%

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

      11. div-inv 96.0%

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

      12. *-commutative 96.0%

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

      13. associate-*r* 91.7%

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

      14. times-frac 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l/ 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-/l/ 84.8%

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

      4. associate-/l/ 84.8%

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

   10. Simplified 84.8%

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

       1. associate-*r/ 88.8%

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

       2. associate-/r* 88.8%

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

       3. frac-times 88.8%

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

       4. associate-/l/ 88.8%

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

       5. div-inv 88.8%

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

       6. associate-/r* 88.8%

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

       7. *-commutative 88.8%

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

       8. associate-*r* 92.4%

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

       9. *-commutative 92.4%

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

   12. Applied egg-rr 92.4%

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

   if 1.12e-281 < c

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. times-frac 70.7%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. div-inv 98.9%

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

      3. div-inv 98.9%

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

      4. div-inv 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. Applied egg-rr 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. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.12 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% 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 70.3%

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

   1. *-un-lft-identity 70.3%

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

   2. add-sqr-sqrt 70.2%

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

   3. times-frac 70.2%

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

4. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. div-inv 97.4%

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

   3. div-inv 97.4%

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

   4. div-inv 97.4%

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

   5. *-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

Alternative 8: 93.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. *-un-lft-identity 70.3%

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

   2. add-sqr-sqrt 70.2%

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

   3. times-frac 70.2%

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

4. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-/r* 97.4%

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

   3. frac-times 94.0%

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

   4. div-inv 94.0%

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

   5. *-commutative 94.0%

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

6. Applied egg-rr 94.0%

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

7. Final simplification 94.0%

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

Alternative 9: 79.2% accurate, 20.9× speedup

Math

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

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))) (/ (/ (/ 1.0 s_m) x_m) c_m)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (1.0 / (c_m * (x_m * s_m))) * (((1.0 / s_m) / x_m) / c_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))) * (((1.0d0 / s_m) / x_m) / c_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))) * (((1.0 / s_m) / x_m) / c_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))) * (((1.0 / s_m) / x_m) / c_m)

Julia

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. *-un-lft-identity 70.3%

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

   2. add-sqr-sqrt 70.2%

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

   3. times-frac 70.2%

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in x around 0 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-/l/ 80.4%

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

   3. associate-/l/ 80.4%

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

7. Simplified 80.4%

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

Alternative 10: 79.1% 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 70.3%

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

3. Taylor expanded in x around 0 60.1%

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

   1. associate-/r* 59.7%

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

   2. *-commutative 59.7%

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

   3. unpow2 59.7%

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

   4. unpow2 59.7%

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

   5. swap-sqr 69.1%

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

   6. unpow2 69.1%

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

   7. associate-/r* 69.5%

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

   8. unpow2 69.5%

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

   9. unpow2 69.5%

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

   10. swap-sqr 80.4%

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

   11. unpow2 80.4%

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

   12. *-commutative 80.4%

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

5. Simplified 80.4%

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

   1. *-commutative 80.4%

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

   2. *-commutative 80.4%

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

   3. *-commutative 80.4%

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

   4. unpow2 80.4%

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

7. Applied egg-rr 80.4%

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

Alternative 11: 75.6% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* (* c_m s_m) (* x_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 * s_m) * (x_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 * s_m) * (x_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 * s_m) * (x_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 * s_m) * (x_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(1.0 / Float64(Float64(c_m * s_m) * Float64(x_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 * s_m) * (x_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[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

3. Taylor expanded in x around 0 60.1%

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

   1. associate-/r* 59.7%

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

   2. *-commutative 59.7%

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

   3. unpow2 59.7%

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

   4. unpow2 59.7%

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

   5. swap-sqr 69.1%

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

   6. unpow2 69.1%

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

   7. associate-/r* 69.5%

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

   8. unpow2 69.5%

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

   9. unpow2 69.5%

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

   10. swap-sqr 80.4%

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

   11. unpow2 80.4%

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

   12. *-commutative 80.4%

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

5. Simplified 80.4%

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

   1. unpow2 80.4%

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

   2. associate-*r* 79.2%

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

   3. *-commutative 79.2%

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

   4. associate-*l* 75.7%

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

7. Applied egg-rr 75.7%

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

Reproduce

herbie shell --seed 2024121 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))