mixedcos

Percentage Accurate: 66.0% → 97.1%

Time: 14.1s

Alternatives: 9

Speedup: 24.1×

• 32.2% 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: 66.0% 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: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 22000:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m} \cdot \left(\frac{1}{c\_m} \cdot \frac{1}{x\_m \cdot s\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{s\_m \cdot \left(\left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

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 22000.0)
   (* (/ (/ 1.0 c_m) (* x_m s_m)) (* (/ 1.0 c_m) (/ 1.0 (* x_m s_m))))
   (/ (cos (* 2.0 x_m)) (* 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 <= 22000.0) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	} else {
		tmp = cos((2.0 * x_m)) / (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 <= 22000.0d0) then
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) * ((1.0d0 / c_m) * (1.0d0 / (x_m * s_m)))
    else
        tmp = cos((2.0d0 * x_m)) / (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 <= 22000.0) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	} else {
		tmp = Math.cos((2.0 * x_m)) / (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 <= 22000.0:
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)))
	else:
		tmp = math.cos((2.0 * x_m)) / (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 <= 22000.0)
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) * Float64(Float64(1.0 / c_m) * Float64(1.0 / Float64(x_m * s_m))));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(s_m * Float64(Float64(x_m * Float64(c_m * 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 <= 22000.0)
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	else
		tmp = cos((2.0 * x_m)) / (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, 22000.0], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 22000

   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-/r* 65.8%

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

      2. cos-neg 65.8%

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

      3. distribute-rgt-neg-out 65.8%

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

      4. distribute-rgt-neg-out 65.8%

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

      5. *-commutative 65.8%

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

      6. distribute-rgt-neg-in 65.8%

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

      7. metadata-eval 65.8%

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

      8. *-commutative 65.8%

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

      9. associate-*l* 60.6%

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

      10. unpow2 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in x around 0 55.4%

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

      1. associate-/r* 55.4%

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

      2. *-commutative 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

      5. swap-sqr 69.3%

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

      6. unpow2 69.3%

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

      7. associate-/r* 69.3%

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

      8. unpow2 69.3%

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

      9. unpow2 69.3%

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

      10. swap-sqr 84.3%

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

      11. unpow2 84.3%

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

      12. *-commutative 84.3%

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

      13. associate-*l* 84.9%

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

   7. Simplified 84.9%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

      3. add-sqr-sqrt 84.2%

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

      4. sqrt-div 84.2%

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

      5. metadata-eval 84.2%

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

      6. *-commutative 84.2%

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

      7. associate-*r* 82.1%

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

      8. unpow2 82.1%

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

      9. sqrt-prod 48.1%

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

      10. add-sqr-sqrt 56.6%

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

      11. *-commutative 56.6%

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

      12. sqrt-div 56.6%

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

      13. metadata-eval 56.6%

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

      14. *-commutative 56.6%

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

      15. associate-*r* 58.5%

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

      16. unpow2 58.5%

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

      17. sqrt-prod 43.7%

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

      18. add-sqr-sqrt 85.0%

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

      19. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. *-commutative 85.0%

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

       2. associate-*r* 82.2%

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

       3. associate-/r* 82.2%

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

       4. div-inv 82.2%

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

       5. *-commutative 82.2%

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

       6. *-commutative 82.2%

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

   11. Applied egg-rr 82.2%

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

   12. Taylor expanded in x around 0 84.4%

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

       1. associate-/r* 84.4%

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

   14. Simplified 84.4%

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

   if 22000 < x

   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. add-sqr-sqrt 72.4%

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

      2. pow2 72.4%

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

      3. sqrt-prod 72.4%

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

      4. sqrt-pow1 81.7%

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

      5. metadata-eval 81.7%

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

      6. pow1 81.7%

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

      7. sqrt-prod 85.3%

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

      8. *-commutative 85.3%

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

      9. sqrt-prod 88.8%

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

      10. sqrt-pow1 99.6%

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

      11. metadata-eval 99.6%

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

      12. pow1 99.6%

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

      13. associate-*r* 99.6%

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

      14. add-sqr-sqrt 99.7%

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

      15. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. associate-*r* 98.1%

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

      5. associate-*r* 96.4%

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

      6. *-commutative 96.4%

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

      7. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 22000:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.1%

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

   2. pow2 67.1%

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

   3. sqrt-prod 67.1%

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

   4. sqrt-pow1 75.1%

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

   5. metadata-eval 75.1%

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

   6. pow1 75.1%

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

   7. sqrt-prod 39.1%

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

   8. *-commutative 39.1%

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

   9. sqrt-prod 40.0%

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

   10. sqrt-pow1 46.6%

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

   11. metadata-eval 46.6%

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

   12. pow1 46.6%

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

   13. associate-*r* 46.6%

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

   14. add-sqr-sqrt 96.6%

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

   15. *-commutative 96.6%

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

4. Applied egg-rr 96.6%

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

   1. div-inv 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*r* 97.4%

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

   4. pow-flip 97.5%

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

   5. *-commutative 97.5%

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

   6. metadata-eval 97.5%

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

6. Applied egg-rr 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r* 96.7%

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

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

Alternative 3: 96.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m} \cdot \left(\frac{1}{c\_m} \cdot \frac{1}{x\_m \cdot s\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 8e-80)
   (* (/ (/ 1.0 c_m) (* x_m s_m)) (* (/ 1.0 c_m) (/ 1.0 (* x_m s_m))))
   (/ (cos (* 2.0 x_m)) (* (* c_m s_m) (* x_m (* x_m (* c_m s_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 8e-80) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	} else {
		tmp = cos((2.0 * x_m)) / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 8d-80) then
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) * ((1.0d0 / c_m) * (1.0d0 / (x_m * s_m)))
    else
        tmp = cos((2.0d0 * x_m)) / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 8e-80) {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	} else {
		tmp = Math.cos((2.0 * x_m)) / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 8e-80:
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)))
	else:
		tmp = math.cos((2.0 * x_m)) / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 8e-80)
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) * Float64(Float64(1.0 / c_m) * Float64(1.0 / Float64(x_m * s_m))));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(x_m * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 8e-80)
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	else
		tmp = cos((2.0 * x_m)) / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 8e-80], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999969e-80

   1. Initial program 62.8%

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

      1. associate-/r* 62.8%

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

      2. cos-neg 62.8%

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

      3. distribute-rgt-neg-out 62.8%

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

      4. distribute-rgt-neg-out 62.8%

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

      5. *-commutative 62.8%

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

      6. distribute-rgt-neg-in 62.8%

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

      7. metadata-eval 62.8%

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

      8. *-commutative 62.8%

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

      9. associate-*l* 56.9%

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

      10. unpow2 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in x around 0 51.0%

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

      1. associate-/r* 51.0%

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

      2. *-commutative 51.0%

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

      3. unpow2 51.0%

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

      4. unpow2 51.0%

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

      5. swap-sqr 66.7%

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

      6. unpow2 66.7%

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

      7. associate-/r* 66.7%

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

      8. unpow2 66.7%

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

      9. unpow2 66.7%

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

      10. swap-sqr 82.2%

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

      11. unpow2 82.2%

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

      12. *-commutative 82.2%

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

      13. associate-*l* 83.0%

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

   7. Simplified 83.0%

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

      1. associate-*r* 82.2%

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

      2. *-commutative 82.2%

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

      3. add-sqr-sqrt 82.2%

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

      4. sqrt-div 82.2%

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

      5. metadata-eval 82.2%

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

      6. *-commutative 82.2%

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

      7. associate-*r* 79.8%

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

      8. unpow2 79.8%

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

      9. sqrt-prod 47.9%

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

      10. add-sqr-sqrt 55.3%

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

      11. *-commutative 55.3%

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

      12. sqrt-div 55.3%

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

      13. metadata-eval 55.3%

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

      14. *-commutative 55.3%

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

      15. associate-*r* 57.4%

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

      16. unpow2 57.4%

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

      17. sqrt-prod 43.4%

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

      18. add-sqr-sqrt 83.1%

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

      19. *-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. *-commutative 83.1%

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

       2. associate-*r* 79.9%

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

       3. associate-/r* 79.9%

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

       4. div-inv 79.9%

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

       5. *-commutative 79.9%

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

       6. *-commutative 79.9%

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

   11. Applied egg-rr 79.9%

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

   12. Taylor expanded in x around 0 82.4%

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

       1. associate-/r* 82.4%

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

   14. Simplified 82.4%

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

   if 7.99999999999999969e-80 < x

   1. Initial program 77.4%

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

      1. add-sqr-sqrt 77.3%

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

      2. pow2 77.3%

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

      3. sqrt-prod 77.4%

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

      4. sqrt-pow1 86.1%

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

      5. metadata-eval 86.1%

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

      6. pow1 86.1%

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

      7. sqrt-prod 88.5%

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

      8. *-commutative 88.5%

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

      9. sqrt-prod 90.9%

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

      10. sqrt-pow1 99.4%

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

      11. metadata-eval 99.4%

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

      12. pow1 99.4%

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

      13. associate-*r* 99.5%

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

      14. add-sqr-sqrt 99.7%

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

      15. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.6%

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

      3. unpow2 99.6%

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

      4. associate-*r* 98.4%

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

      5. *-commutative 98.4%

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

      6. *-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 5.3 \cdot 10^{+194}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{x\_m \cdot s\_m} \cdot \left(\frac{1}{c\_m} \cdot \frac{1}{x\_m \cdot s\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= s_m 5.3e+194)
   (/ (cos (* 2.0 x_m)) (* (* x_m c_m) (* s_m (* x_m (* c_m s_m)))))
   (* (/ (/ 1.0 c_m) (* x_m s_m)) (* (/ 1.0 c_m) (/ 1.0 (* x_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 5.3e+194) {
		tmp = cos((2.0 * x_m)) / ((x_m * c_m) * (s_m * (x_m * (c_m * s_m))));
	} else {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (s_m <= 5.3d+194) then
        tmp = cos((2.0d0 * x_m)) / ((x_m * c_m) * (s_m * (x_m * (c_m * s_m))))
    else
        tmp = ((1.0d0 / c_m) / (x_m * s_m)) * ((1.0d0 / c_m) * (1.0d0 / (x_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 5.3e+194) {
		tmp = Math.cos((2.0 * x_m)) / ((x_m * c_m) * (s_m * (x_m * (c_m * s_m))));
	} else {
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if s_m <= 5.3e+194:
		tmp = math.cos((2.0 * x_m)) / ((x_m * c_m) * (s_m * (x_m * (c_m * s_m))))
	else:
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (s_m <= 5.3e+194)
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * Float64(c_m * s_m)))));
	else
		tmp = Float64(Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)) * Float64(Float64(1.0 / c_m) * Float64(1.0 / Float64(x_m * s_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (s_m <= 5.3e+194)
		tmp = cos((2.0 * x_m)) / ((x_m * c_m) * (s_m * (x_m * (c_m * s_m))));
	else
		tmp = ((1.0 / c_m) / (x_m * s_m)) * ((1.0 / c_m) * (1.0 / (x_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[s$95$m, 5.3e+194], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5.30000000000000005e194

   1. Initial program 66.7%

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

      1. add-sqr-sqrt 66.7%

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

      2. pow2 66.7%

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

      3. sqrt-prod 66.7%

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

      4. sqrt-pow1 74.5%

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

      5. metadata-eval 74.5%

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

      6. pow1 74.5%

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

      7. sqrt-prod 38.7%

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

      8. *-commutative 38.7%

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

      9. sqrt-prod 39.6%

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

      10. sqrt-pow1 45.7%

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

      11. metadata-eval 45.7%

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

      12. pow1 45.7%

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

      13. associate-*r* 45.7%

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

      14. add-sqr-sqrt 96.4%

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

      15. *-commutative 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. unpow2 96.4%

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

      2. associate-*r* 95.3%

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

      3. *-commutative 95.3%

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

      4. associate-*r* 96.9%

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

      5. associate-*l* 93.0%

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

      6. *-commutative 93.0%

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

      7. *-commutative 93.0%

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

   6. Applied egg-rr 93.0%

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

   if 5.30000000000000005e194 < s

   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-/r* 72.4%

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

      2. cos-neg 72.4%

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

      3. distribute-rgt-neg-out 72.4%

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

      4. distribute-rgt-neg-out 72.4%

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

      5. *-commutative 72.4%

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

      6. distribute-rgt-neg-in 72.4%

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

      7. metadata-eval 72.4%

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

      8. *-commutative 72.4%

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

      9. associate-*l* 66.6%

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

      10. unpow2 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. associate-/r* 66.6%

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

      2. *-commutative 66.6%

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

      3. unpow2 66.6%

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

      4. unpow2 66.6%

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

      5. swap-sqr 86.0%

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

      6. unpow2 86.0%

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

      7. associate-/r* 85.8%

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

      8. unpow2 85.8%

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

      9. unpow2 85.8%

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

      10. swap-sqr 99.6%

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

      11. unpow2 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-*l* 91.1%

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

   7. Simplified 91.1%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 99.5%

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

      4. sqrt-div 99.5%

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

      5. metadata-eval 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 91.1%

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

      8. unpow2 91.1%

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

      9. sqrt-prod 52.4%

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

      10. add-sqr-sqrt 81.8%

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

      11. *-commutative 81.8%

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

      12. sqrt-div 81.8%

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

      13. metadata-eval 81.8%

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

      14. *-commutative 81.8%

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

      15. associate-*r* 81.8%

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

      16. unpow2 81.8%

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

      17. sqrt-prod 52.4%

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

      18. add-sqr-sqrt 91.2%

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

      19. *-commutative 91.2%

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

   9. Applied egg-rr 91.2%

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

       1. *-commutative 91.2%

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

       2. associate-*r* 91.2%

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

       3. associate-/r* 91.2%

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

       4. div-inv 91.3%

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

       5. *-commutative 91.3%

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

       6. *-commutative 91.3%

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

   11. Applied egg-rr 91.3%

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

   12. Taylor expanded in x around 0 99.9%

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

       1. associate-/r* 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.3 \cdot 10^{+194}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \left(\frac{1}{c} \cdot \frac{1}{x \cdot s}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 2.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m)))) (/ (/ (cos (* 2.0 x_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 = c_m * (x_m * s_m);
	return (cos((2.0 * x_m)) / t_0) / t_0;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.1%

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

   2. pow2 67.1%

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

   3. sqrt-prod 67.1%

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

   4. sqrt-pow1 75.1%

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

   5. metadata-eval 75.1%

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

   6. pow1 75.1%

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

   7. sqrt-prod 39.1%

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

   8. *-commutative 39.1%

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

   9. sqrt-prod 40.0%

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

   10. sqrt-pow1 46.6%

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

   11. metadata-eval 46.6%

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

   12. pow1 46.6%

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

   13. associate-*r* 46.6%

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

   14. add-sqr-sqrt 96.6%

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

   15. *-commutative 96.6%

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

4. Applied egg-rr 96.6%

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

   1. div-inv 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*r* 97.4%

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

   4. pow-flip 97.5%

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

   5. *-commutative 97.5%

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

   6. metadata-eval 97.5%

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

6. Applied egg-rr 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r* 96.7%

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

8. Simplified 96.7%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

   3. metadata-eval 97.5%

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

   4. pow-div 97.4%

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

   5. inv-pow 97.4%

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

   6. pow1 97.4%

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

   7. associate-*r/ 97.4%

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

   8. un-div-inv 97.4%

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

   9. *-commutative 97.4%

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

   10. associate-*r* 94.6%

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

   11. *-commutative 94.6%

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

   12. *-commutative 94.6%

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

   13. associate-*r* 96.7%

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

   14. *-commutative 96.7%

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

10. Applied egg-rr 96.7%

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

Alternative 6: 79.5% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. associate-/r* 67.2%

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

   2. cos-neg 67.2%

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

   3. distribute-rgt-neg-out 67.2%

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

   4. distribute-rgt-neg-out 67.2%

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

   5. *-commutative 67.2%

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

   6. distribute-rgt-neg-in 67.2%

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

   7. metadata-eval 67.2%

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

   8. *-commutative 67.2%

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

   9. associate-*l* 60.7%

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

   10. unpow2 60.7%

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

3. Simplified 60.7%

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

5. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.6%

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

   2. *-commutative 55.6%

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

   3. unpow2 55.6%

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

   4. unpow2 55.6%

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

   5. swap-sqr 68.3%

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

   6. unpow2 68.3%

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

   7. associate-/r* 68.3%

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

   8. unpow2 68.3%

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

   9. unpow2 68.3%

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

   10. swap-sqr 81.3%

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

   11. unpow2 81.3%

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

   12. *-commutative 81.3%

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

   13. associate-*l* 81.9%

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

7. Simplified 81.9%

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

   1. associate-*r* 81.3%

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

   2. *-commutative 81.3%

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

   3. add-sqr-sqrt 81.3%

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

   4. sqrt-div 81.3%

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

   5. metadata-eval 81.3%

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

   6. *-commutative 81.3%

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

   7. associate-*r* 79.6%

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

   8. unpow2 79.6%

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

   9. sqrt-prod 47.3%

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

   10. add-sqr-sqrt 59.4%

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

   11. *-commutative 59.4%

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

   12. sqrt-div 59.4%

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

   13. metadata-eval 59.4%

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

   14. *-commutative 59.4%

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

   15. associate-*r* 60.9%

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

   16. unpow2 60.9%

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

   17. sqrt-prod 44.2%

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

   18. add-sqr-sqrt 81.9%

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

   19. *-commutative 81.9%

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

9. Applied egg-rr 81.9%

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

    1. *-commutative 81.9%

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

    2. associate-*r* 79.7%

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

    3. associate-/r* 79.7%

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

    4. div-inv 79.7%

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

    5. *-commutative 79.7%

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

    6. *-commutative 79.7%

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

11. Applied egg-rr 79.7%

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

12. Taylor expanded in x around 0 81.4%

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

    1. associate-/r* 81.4%

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

14. Simplified 81.4%

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

15. Final simplification 81.4%

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

Alternative 7: 79.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.1%

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

   2. pow2 67.1%

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

   3. sqrt-prod 67.1%

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

   4. sqrt-pow1 75.1%

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

   5. metadata-eval 75.1%

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

   6. pow1 75.1%

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

   7. sqrt-prod 39.1%

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

   8. *-commutative 39.1%

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

   9. sqrt-prod 40.0%

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

   10. sqrt-pow1 46.6%

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

   11. metadata-eval 46.6%

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

   12. pow1 46.6%

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

   13. associate-*r* 46.6%

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

   14. add-sqr-sqrt 96.6%

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

   15. *-commutative 96.6%

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

4. Applied egg-rr 96.6%

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

   1. div-inv 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*r* 97.4%

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

   4. pow-flip 97.5%

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

   5. *-commutative 97.5%

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

   6. metadata-eval 97.5%

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

6. Applied egg-rr 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r* 96.7%

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

8. Simplified 96.7%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

   3. metadata-eval 97.5%

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

   4. pow-div 97.4%

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

   5. inv-pow 97.4%

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

   6. pow1 97.4%

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

   7. associate-*r/ 97.4%

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

   8. un-div-inv 97.4%

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

   9. *-commutative 97.4%

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

   10. associate-*r* 94.6%

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

   11. *-commutative 94.6%

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

   12. *-commutative 94.6%

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

   13. associate-*r* 96.7%

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

   14. *-commutative 96.7%

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

10. Applied egg-rr 96.7%

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

11. Taylor expanded in x around 0 81.4%

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

    1. associate-/r* 81.4%

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

13. Simplified 81.4%

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

14. Final simplification 81.4%

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

Alternative 8: 79.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.1%

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

   2. pow2 67.1%

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

   3. sqrt-prod 67.1%

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

   4. sqrt-pow1 75.1%

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

   5. metadata-eval 75.1%

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

   6. pow1 75.1%

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

   7. sqrt-prod 39.1%

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

   8. *-commutative 39.1%

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

   9. sqrt-prod 40.0%

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

   10. sqrt-pow1 46.6%

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

   11. metadata-eval 46.6%

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

   12. pow1 46.6%

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

   13. associate-*r* 46.6%

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

   14. add-sqr-sqrt 96.6%

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

   15. *-commutative 96.6%

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

4. Applied egg-rr 96.6%

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

   1. div-inv 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*r* 97.4%

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

   4. pow-flip 97.5%

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

   5. *-commutative 97.5%

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

   6. metadata-eval 97.5%

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

6. Applied egg-rr 97.5%

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

   1. *-commutative 97.5%

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

   2. associate-*r* 96.7%

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

8. Simplified 96.7%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

   3. metadata-eval 97.5%

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

   4. pow-div 97.4%

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

   5. inv-pow 97.4%

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

   6. pow1 97.4%

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

   7. associate-*r/ 97.4%

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

   8. un-div-inv 97.4%

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

   9. *-commutative 97.4%

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

   10. associate-*r* 94.6%

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

   11. *-commutative 94.6%

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

   12. *-commutative 94.6%

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

   13. associate-*r* 96.7%

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

   14. *-commutative 96.7%

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

10. Applied egg-rr 96.7%

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

11. Taylor expanded in x around 0 81.4%

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

Alternative 9: 76.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. add-sqr-sqrt 67.1%

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

   2. pow2 67.1%

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

   3. sqrt-prod 67.1%

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

   4. sqrt-pow1 75.1%

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

   5. metadata-eval 75.1%

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

   6. pow1 75.1%

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

   7. sqrt-prod 39.1%

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

   8. *-commutative 39.1%

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

   9. sqrt-prod 40.0%

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

   10. sqrt-pow1 46.6%

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

   11. metadata-eval 46.6%

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

   12. pow1 46.6%

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

   13. associate-*r* 46.6%

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

   14. add-sqr-sqrt 96.6%

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

   15. *-commutative 96.6%

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

4. Applied egg-rr 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*r* 97.4%

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

   3. unpow2 97.4%

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

   4. associate-*r* 94.8%

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

   5. *-commutative 94.8%

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

   6. *-commutative 94.8%

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

6. Applied egg-rr 94.8%

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

7. Taylor expanded in x around 0 80.0%

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

8. Final simplification 80.0%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))