mixedcos

Percentage Accurate: 66.4% → 98.9%

Time: 15.3s

Alternatives: 11

Speedup: 24.1×

• 31.6% 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: 98.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{x\_m \cdot s\_m} \cdot \frac{-1}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{s\_m}}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 2.95e-8)
   (/ (* (/ -1.0 (* x_m s_m)) (/ -1.0 c_m)) (* (* x_m s_m) c_m))
   (/ (/ (cos (* x_m 2.0)) s_m) (* (* x_m c_m) (* s_m (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.95e-8) {
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_m) * c_m);
	} else {
		tmp = (cos((x_m * 2.0)) / s_m) / ((x_m * c_m) * (s_m * (x_m * c_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.95d-8) then
        tmp = (((-1.0d0) / (x_m * s_m)) * ((-1.0d0) / c_m)) / ((x_m * s_m) * c_m)
    else
        tmp = (cos((x_m * 2.0d0)) / s_m) / ((x_m * c_m) * (s_m * (x_m * c_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.95e-8) {
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_m) * c_m);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / s_m) / ((x_m * c_m) * (s_m * (x_m * c_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2.95e-8:
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_m) * c_m)
	else:
		tmp = (math.cos((x_m * 2.0)) / s_m) / ((x_m * c_m) * (s_m * (x_m * c_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2.95e-8)
		tmp = Float64(Float64(Float64(-1.0 / Float64(x_m * s_m)) * Float64(-1.0 / c_m)) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / s_m) / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2.95e-8)
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_m) * c_m);
	else
		tmp = (cos((x_m * 2.0)) / s_m) / ((x_m * c_m) * (s_m * (x_m * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2.95e-8], N[(N[(N[(-1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9499999999999999e-8

   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. Add Preprocessing

   3. Taylor expanded in x around 0 65.1%

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

      1. associate-/r* 65.1%

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

      2. *-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. swap-sqr 78.6%

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

      6. unpow2 78.6%

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

      7. associate-/r* 79.0%

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

      8. unpow2 79.0%

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

      9. unpow2 79.0%

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

      10. swap-sqr 88.5%

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

      11. unpow2 88.5%

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

      12. *-commutative 88.5%

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

   5. Simplified 88.5%

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

      1. *-commutative 88.5%

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

      2. metadata-eval 88.5%

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

      3. unpow2 88.5%

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

      4. frac-times 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. associate-/l/ 88.7%

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

      2. associate-/r* 88.7%

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

      3. frac-2neg 88.7%

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

      4. frac-times 88.7%

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

      5. distribute-neg-frac 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

   if 2.9499999999999999e-8 < x

   1. Initial program 65.8%

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

      1. associate-/l/ 65.8%

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

      2. remove-double-neg 65.8%

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

      3. distribute-frac-neg 65.8%

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

      4. distribute-neg-frac 65.8%

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

      5. remove-double-neg 65.8%

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

      6. *-commutative 65.8%

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

      7. associate-*r* 60.1%

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

      8. unpow2 60.1%

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

      9. associate-/r* 60.0%

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

      10. cos-neg 60.0%

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

      11. *-commutative 60.0%

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

      12. distribute-rgt-neg-in 60.0%

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

      13. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

      1. associate-/l/ 60.1%

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

      2. *-un-lft-identity 60.1%

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

      3. add-sqr-sqrt 60.1%

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

      4. times-frac 60.1%

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

      5. pow-prod-down 60.1%

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

      6. sqrt-pow1 47.7%

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

      7. metadata-eval 47.7%

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

      8. pow1 47.7%

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

      9. *-commutative 47.7%

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

      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 22.4%

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

      12. *-commutative 22.4%

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

      13. *-commutative 22.4%

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

      14. swap-sqr 22.4%

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

      15. metadata-eval 22.4%

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

      16. metadata-eval 22.4%

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

      17. swap-sqr 22.4%

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

      18. sqrt-unprod 47.8%

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

      19. add-sqr-sqrt 47.7%

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

   6. Applied egg-rr 75.4%

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

      1. div-inv 74.0%

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

      2. frac-times 73.7%

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

      3. *-un-lft-identity 73.7%

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

      4. pow2 73.7%

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

      5. frac-times 75.1%

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

      6. unpow-prod-down 97.2%

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

      7. *-commutative 97.2%

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

      8. unpow2 97.2%

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

      9. frac-times 97.9%

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

      10. div-inv 97.9%

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

      11. *-un-lft-identity 97.9%

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

      12. associate-*r* 96.6%

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

      13. times-frac 92.8%

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

      14. *-commutative 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-/r* 94.0%

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

      2. associate-/l/ 94.0%

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

      3. *-commutative 94.0%

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

   10. Simplified 94.0%

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

       1. clear-num 92.8%

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

       2. associate-/r* 92.8%

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

       3. *-commutative 92.8%

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

       4. frac-times 94.6%

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

       5. *-un-lft-identity 94.6%

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

       6. div-inv 94.6%

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

       7. clear-num 94.6%

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

       8. /-rgt-identity 94.6%

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

       9. associate-*r* 96.2%

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

   12. Applied egg-rr 96.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot s} \cdot \frac{-1}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 2.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* x_m s_m) c_m)))
   (if (<= x_m 1e-7)
     (/ (* (/ -1.0 (* x_m s_m)) (/ -1.0 c_m)) t_0)
     (/ (/ (cos (* x_m 2.0)) s_m) (* t_0 (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	double tmp;
	if (x_m <= 1e-7) {
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / t_0;
	} else {
		tmp = (cos((x_m * 2.0)) / s_m) / (t_0 * (x_m * c_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * s_m) * c_m
    if (x_m <= 1d-7) then
        tmp = (((-1.0d0) / (x_m * s_m)) * ((-1.0d0) / c_m)) / t_0
    else
        tmp = (cos((x_m * 2.0d0)) / s_m) / (t_0 * (x_m * c_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	double tmp;
	if (x_m <= 1e-7) {
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / t_0;
	} else {
		tmp = (Math.cos((x_m * 2.0)) / s_m) / (t_0 * (x_m * c_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (x_m * s_m) * c_m
	tmp = 0
	if x_m <= 1e-7:
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / t_0
	else:
		tmp = (math.cos((x_m * 2.0)) / s_m) / (t_0 * (x_m * c_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(x_m * s_m) * c_m)
	tmp = 0.0
	if (x_m <= 1e-7)
		tmp = Float64(Float64(Float64(-1.0 / Float64(x_m * s_m)) * Float64(-1.0 / c_m)) / t_0);
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / s_m) / Float64(t_0 * Float64(x_m * c_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (x_m * s_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1e-7)
		tmp = ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / t_0;
	else
		tmp = (cos((x_m * 2.0)) / s_m) / (t_0 * (x_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-8

   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. Add Preprocessing

   3. Taylor expanded in x around 0 65.1%

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

      1. associate-/r* 65.1%

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

      2. *-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. swap-sqr 78.6%

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

      6. unpow2 78.6%

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

      7. associate-/r* 79.0%

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

      8. unpow2 79.0%

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

      9. unpow2 79.0%

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

      10. swap-sqr 88.5%

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

      11. unpow2 88.5%

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

      12. *-commutative 88.5%

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

   5. Simplified 88.5%

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

      1. *-commutative 88.5%

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

      2. metadata-eval 88.5%

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

      3. unpow2 88.5%

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

      4. frac-times 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. associate-/l/ 88.7%

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

      2. associate-/r* 88.7%

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

      3. frac-2neg 88.7%

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

      4. frac-times 88.7%

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

      5. distribute-neg-frac 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 65.8%

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

      1. associate-/l/ 65.8%

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

      2. remove-double-neg 65.8%

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

      3. distribute-frac-neg 65.8%

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

      4. distribute-neg-frac 65.8%

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

      5. remove-double-neg 65.8%

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

      6. *-commutative 65.8%

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

      7. associate-*r* 60.1%

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

      8. unpow2 60.1%

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

      9. associate-/r* 60.0%

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

      10. cos-neg 60.0%

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

      11. *-commutative 60.0%

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

      12. distribute-rgt-neg-in 60.0%

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

      13. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

      1. associate-/l/ 60.1%

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

      2. *-un-lft-identity 60.1%

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

      3. add-sqr-sqrt 60.1%

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

      4. times-frac 60.1%

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

      5. pow-prod-down 60.1%

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

      6. sqrt-pow1 47.7%

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

      7. metadata-eval 47.7%

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

      8. pow1 47.7%

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

      9. *-commutative 47.7%

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

      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 22.4%

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

      12. *-commutative 22.4%

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

      13. *-commutative 22.4%

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

      14. swap-sqr 22.4%

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

      15. metadata-eval 22.4%

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

      16. metadata-eval 22.4%

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

      17. swap-sqr 22.4%

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

      18. sqrt-unprod 47.8%

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

      19. add-sqr-sqrt 47.7%

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

   6. Applied egg-rr 75.4%

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

      1. div-inv 74.0%

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

      2. frac-times 73.7%

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

      3. *-un-lft-identity 73.7%

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

      4. pow2 73.7%

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

      5. frac-times 75.1%

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

      6. unpow-prod-down 97.2%

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

      7. *-commutative 97.2%

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

      8. unpow2 97.2%

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

      9. frac-times 97.9%

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

      10. div-inv 97.9%

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

      11. *-un-lft-identity 97.9%

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

      12. associate-*r* 96.6%

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

      13. times-frac 92.8%

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

      14. *-commutative 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-/r* 94.0%

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

      2. associate-/l/ 94.0%

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

      3. *-commutative 94.0%

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

   10. Simplified 94.0%

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

       1. clear-num 92.8%

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

       2. associate-/r* 92.8%

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

       3. *-commutative 92.8%

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

       4. frac-times 94.6%

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

       5. *-un-lft-identity 94.6%

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

       6. div-inv 94.6%

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

       7. clear-num 94.6%

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

       8. /-rgt-identity 94.6%

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

       9. associate-*r* 96.2%

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

   12. Applied egg-rr 96.2%

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

   13. Taylor expanded in c around 0 94.6%

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

4. Final simplification 90.4%

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

Alternative 3: 96.8% accurate, 2.7× speedup

Math

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

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
 (* (/ (/ (cos (* x_m 2.0)) (* x_m s_m)) c_m) (/ (/ 1.0 (* x_m s_m)) c_m)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return ((cos((x_m * 2.0)) / (x_m * s_m)) / c_m) * ((1.0 / (x_m * s_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 = ((cos((x_m * 2.0d0)) / (x_m * s_m)) / c_m) * ((1.0d0 / (x_m * s_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 ((Math.cos((x_m * 2.0)) / (x_m * s_m)) / c_m) * ((1.0 / (x_m * s_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 ((math.cos((x_m * 2.0)) / (x_m * s_m)) / c_m) * ((1.0 / (x_m * s_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(Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * s_m)) / c_m) * Float64(Float64(1.0 / Float64(x_m * s_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 = ((cos((x_m * 2.0)) / (x_m * s_m)) / c_m) * ((1.0 / (x_m * s_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[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

   1. associate-/l/ 72.4%

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

   2. remove-double-neg 72.4%

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

   3. distribute-frac-neg 72.4%

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

   4. distribute-neg-frac 72.4%

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

   5. remove-double-neg 72.4%

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

   6. *-commutative 72.4%

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

   7. associate-*r* 66.5%

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

   8. unpow2 66.5%

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

   9. associate-/r* 66.2%

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

   10. cos-neg 66.2%

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

   11. *-commutative 66.2%

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

   12. distribute-rgt-neg-in 66.2%

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

   13. metadata-eval 66.2%

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

3. Simplified 66.2%

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

   1. associate-/l/ 66.5%

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

   2. *-un-lft-identity 66.5%

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

   3. add-sqr-sqrt 66.4%

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

   4. times-frac 66.4%

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

   5. pow-prod-down 66.4%

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

   6. sqrt-pow1 47.7%

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

   7. metadata-eval 47.7%

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

   8. pow1 47.7%

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

   9. *-commutative 47.7%

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

   10. add-sqr-sqrt 26.8%

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

   11. sqrt-unprod 33.3%

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

   12. *-commutative 33.3%

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

   13. *-commutative 33.3%

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

   14. swap-sqr 33.3%

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

   15. metadata-eval 33.3%

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

   16. metadata-eval 33.3%

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

   17. swap-sqr 33.3%

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

   18. sqrt-unprod 20.5%

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

   19. add-sqr-sqrt 47.7%

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

6. Applied egg-rr 83.1%

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

   1. *-commutative 83.1%

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

   2. unpow2 83.1%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 93.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_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 = (cos((x_m * 2.0d0)) / c_m) / ((x_m * s_m) * ((x_m * s_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 (Math.cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_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 (math.cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_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(cos(Float64(x_m * 2.0)) / c_m) / Float64(Float64(x_m * s_m) * Float64(Float64(x_m * s_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 = (cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_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[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

   1. *-un-lft-identity 72.4%

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

   2. add-sqr-sqrt 72.4%

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

   3. times-frac 72.4%

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

4. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-/r* 98.5%

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

   3. frac-times 96.1%

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

   4. div-inv 96.1%

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

   5. *-commutative 96.1%

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

6. Applied egg-rr 96.1%

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

7. Final simplification 96.1%

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

Alternative 5: 96.8% accurate, 2.7× speedup

Math

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

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 (* (* x_m s_m) c_m))) (/ (/ (cos (* x_m 2.0)) t_0) t_0)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	return (cos((x_m * 2.0)) / t_0) / t_0;
}

Fortran

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

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	return (Math.cos((x_m * 2.0)) / t_0) / t_0;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (x_m * s_m) * c_m
	return (math.cos((x_m * 2.0)) / t_0) / t_0

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(x_m * s_m) * c_m)
	return Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = (x_m * s_m) * c_m;
	tmp = (cos((x_m * 2.0)) / t_0) / t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $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 72.4%

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

   1. *-un-lft-identity 72.4%

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

   2. add-sqr-sqrt 72.4%

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

   3. times-frac 72.4%

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

4. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. div-inv 98.5%

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

   3. div-inv 98.5%

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

   4. div-inv 98.5%

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

   5. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 6: 79.1% accurate, 20.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / ((x_m * s_m) * c_m);
	return t_0 * t_0;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = 1.0d0 / ((x_m * s_m) * c_m)
    code = t_0 * t_0
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / ((x_m * s_m) * c_m);
	return t_0 * t_0;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = 1.0 / ((x_m * s_m) * c_m)
	return t_0 * t_0

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(1.0 / Float64(Float64(x_m * s_m) * c_m))
	return Float64(t_0 * t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = 1.0 / ((x_m * s_m) * c_m);
	tmp = t_0 * t_0;
end

Wolfram

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

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in x around 0 61.0%

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

   1. associate-/r* 60.6%

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

   2. *-commutative 60.6%

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

   3. unpow2 60.6%

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

   4. unpow2 60.6%

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

   5. swap-sqr 71.9%

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

   6. unpow2 71.9%

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

   7. associate-/r* 72.6%

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

   8. unpow2 72.6%

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

   9. unpow2 72.6%

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

   10. swap-sqr 80.6%

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

   11. unpow2 80.6%

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

   12. *-commutative 80.6%

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

5. Simplified 80.6%

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

   1. *-commutative 80.6%

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

   2. metadata-eval 80.6%

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

   3. unpow2 80.6%

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

   4. frac-times 80.7%

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

7. Applied egg-rr 80.7%

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

8. Final simplification 80.7%

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

Alternative 7: 79.1% accurate, 20.9× speedup

Math

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

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 (* x_m s_m)) c_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 / (x_m * s_m)) / c_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 / (x_m * s_m)) / c_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 / (x_m * s_m)) / c_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 / (x_m * s_m)) / c_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(Float64(1.0 / Float64(x_m * s_m)) / c_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 / (x_m * s_m)) / c_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[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(N[(N[(1.0 / s$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

   1. associate-/l/ 72.4%

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

   2. remove-double-neg 72.4%

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

   3. distribute-frac-neg 72.4%

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

   4. distribute-neg-frac 72.4%

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

   5. remove-double-neg 72.4%

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

   6. *-commutative 72.4%

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

   7. associate-*r* 66.5%

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

   8. unpow2 66.5%

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

   9. associate-/r* 66.2%

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

   10. cos-neg 66.2%

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

   11. *-commutative 66.2%

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

   12. distribute-rgt-neg-in 66.2%

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

   13. metadata-eval 66.2%

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

3. Simplified 66.2%

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

   1. associate-/l/ 66.5%

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

   2. *-un-lft-identity 66.5%

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

   3. add-sqr-sqrt 66.4%

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

   4. times-frac 66.4%

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

   5. pow-prod-down 66.4%

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

   6. sqrt-pow1 47.7%

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

   7. metadata-eval 47.7%

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

   8. pow1 47.7%

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

   9. *-commutative 47.7%

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

   10. add-sqr-sqrt 26.8%

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

   11. sqrt-unprod 33.3%

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

   12. *-commutative 33.3%

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

   13. *-commutative 33.3%

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

   14. swap-sqr 33.3%

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

   15. metadata-eval 33.3%

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

   16. metadata-eval 33.3%

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

   17. swap-sqr 33.3%

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

   18. sqrt-unprod 20.5%

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

   19. add-sqr-sqrt 47.7%

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

6. Applied egg-rr 83.1%

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

   1. *-commutative 83.1%

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

   2. unpow2 83.1%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around 0 80.7%

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

    1. associate-/r* 80.7%

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

11. Simplified 80.7%

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

12. Final simplification 80.7%

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

Alternative 8: 79.1% accurate, 20.9× speedup

Math

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

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 (* x_m s_m)) (/ -1.0 c_m)) (* (* x_m s_m) c_m)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return ((-1.0 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_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) / (x_m * s_m)) * ((-1.0d0) / c_m)) / ((x_m * s_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 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_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 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_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(Float64(-1.0 / Float64(x_m * s_m)) * Float64(-1.0 / c_m)) / Float64(Float64(x_m * s_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 / (x_m * s_m)) * (-1.0 / c_m)) / ((x_m * s_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[(N[(-1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in x around 0 61.0%

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

   1. associate-/r* 60.6%

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

   2. *-commutative 60.6%

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

   3. unpow2 60.6%

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

   4. unpow2 60.6%

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

   5. swap-sqr 71.9%

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

   6. unpow2 71.9%

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

   7. associate-/r* 72.6%

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

   8. unpow2 72.6%

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

   9. unpow2 72.6%

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

   10. swap-sqr 80.6%

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

   11. unpow2 80.6%

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

   12. *-commutative 80.6%

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

5. Simplified 80.6%

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

   1. *-commutative 80.6%

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

   2. metadata-eval 80.6%

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

   3. unpow2 80.6%

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

   4. frac-times 80.7%

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

7. Applied egg-rr 80.7%

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

   1. associate-/l/ 80.7%

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

   2. associate-/r* 80.7%

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

   3. frac-2neg 80.7%

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

   4. frac-times 80.7%

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

   5. distribute-neg-frac 80.7%

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

   6. metadata-eval 80.7%

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

9. Applied egg-rr 80.7%

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

10. Final simplification 80.7%

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

Alternative 9: 69.1% accurate, 24.1× speedup

Math

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

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 (* (* s_m c_m) (* s_m (* x_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 / ((s_m * c_m) * (s_m * (x_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 / ((s_m * c_m) * (s_m * (x_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 / ((s_m * c_m) * (s_m * (x_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 / ((s_m * c_m) * (s_m * (x_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(1.0 / Float64(Float64(s_m * c_m) * Float64(s_m * Float64(x_m * Float64(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 / ((s_m * c_m) * (s_m * (x_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[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in x around 0 61.0%

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

   1. associate-/r* 60.6%

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

   2. *-commutative 60.6%

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

   3. unpow2 60.6%

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

   4. unpow2 60.6%

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

   5. swap-sqr 71.9%

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

   6. unpow2 71.9%

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

   7. associate-/r* 72.6%

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

   8. unpow2 72.6%

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

   9. unpow2 72.6%

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

   10. swap-sqr 80.6%

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

   11. unpow2 80.6%

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

   12. *-commutative 80.6%

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

5. Simplified 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*r* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 77.6%

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

7. Applied egg-rr 77.6%

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

   1. pow1 77.6%

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

   2. associate-*r* 76.5%

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

   3. associate-*r* 75.4%

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

9. Applied egg-rr 75.4%

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

    1. unpow1 75.4%

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

    2. *-commutative 75.4%

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

    3. associate-*l* 75.4%

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

11. Simplified 75.4%

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

12. Final simplification 75.4%

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

Alternative 10: 75.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((s_m * c_m) * (x_m * ((x_m * s_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 / ((s_m * c_m) * (x_m * ((x_m * s_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 / ((s_m * c_m) * (x_m * ((x_m * s_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 / ((s_m * c_m) * (x_m * ((x_m * s_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(1.0 / Float64(Float64(s_m * c_m) * Float64(x_m * Float64(Float64(x_m * s_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 / ((s_m * c_m) * (x_m * ((x_m * s_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[(1.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in x around 0 61.0%

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

   1. associate-/r* 60.6%

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

   2. *-commutative 60.6%

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

   3. unpow2 60.6%

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

   4. unpow2 60.6%

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

   5. swap-sqr 71.9%

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

   6. unpow2 71.9%

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

   7. associate-/r* 72.6%

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

   8. unpow2 72.6%

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

   9. unpow2 72.6%

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

   10. swap-sqr 80.6%

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

   11. unpow2 80.6%

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

   12. *-commutative 80.6%

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

5. Simplified 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*r* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 77.6%

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

7. Applied egg-rr 77.6%

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

8. Final simplification 77.6%

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

Alternative 11: 79.0% accurate, 24.1× speedup

Math

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

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 (* (* x_m s_m) c_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 = (x_m * s_m) * c_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 = (x_m * s_m) * c_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 = (x_m * s_m) * c_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 = (x_m * s_m) * c_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(Float64(x_m * s_m) * c_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 = (x_m * s_m) * c_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[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in x around 0 61.0%

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

   1. associate-/r* 60.6%

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

   2. *-commutative 60.6%

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

   3. unpow2 60.6%

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

   4. unpow2 60.6%

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

   5. swap-sqr 71.9%

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

   6. unpow2 71.9%

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

   7. associate-/r* 72.6%

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

   8. unpow2 72.6%

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

   9. unpow2 72.6%

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

   10. swap-sqr 80.6%

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

   11. unpow2 80.6%

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

   12. *-commutative 80.6%

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

5. Simplified 80.6%

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

   1. *-commutative 80.6%

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

   2. unpow2 80.6%

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

7. Applied egg-rr 80.6%

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

8. Final simplification 80.6%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))