mixedcos

Percentage Accurate: 66.9% → 99.6%

Time: 14.7s

Alternatives: 6

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s\_m \cdot \left(x\_m \cdot c\_m\right)} \cdot \frac{\frac{\frac{\cos \left(x\_m \cdot 2\right)}{c\_m}}{x\_m}}{s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 2e-46)
   (/ (/ (/ 1.0 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))
   (* (/ 1.0 (* s_m (* x_m c_m))) (/ (/ (/ (cos (* x_m 2.0)) c_m) x_m) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2e-46) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (1.0 / (s_m * (x_m * c_m))) * (((cos((x_m * 2.0)) / c_m) / x_m) / s_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2d-46) then
        tmp = ((1.0d0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = (1.0d0 / (s_m * (x_m * c_m))) * (((cos((x_m * 2.0d0)) / c_m) / x_m) / s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2e-46) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (1.0 / (s_m * (x_m * c_m))) * (((Math.cos((x_m * 2.0)) / c_m) / x_m) / s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2e-46:
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = (1.0 / (s_m * (x_m * c_m))) * (((math.cos((x_m * 2.0)) / c_m) / x_m) / s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2e-46)
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(Float64(1.0 / Float64(s_m * Float64(x_m * c_m))) * Float64(Float64(Float64(cos(Float64(x_m * 2.0)) / c_m) / x_m) / s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2e-46)
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = (1.0 / (s_m * (x_m * c_m))) * (((cos((x_m * 2.0)) / c_m) / x_m) / s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2e-46], N[(N[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000005e-46

   1. Initial program 71.9%

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

      1. associate-/r* 71.9%

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

      2. *-commutative 71.9%

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

      3. unpow2 71.9%

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

      4. sqr-neg 71.9%

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

      5. unpow2 71.9%

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

      6. cos-neg 71.9%

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

      7. *-commutative 71.9%

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

      8. distribute-rgt-neg-in 71.9%

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

      9. metadata-eval 71.9%

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

      10. unpow2 71.9%

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

      11. sqr-neg 71.9%

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

      12. unpow2 71.9%

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

      13. associate-*r* 66.8%

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

      14. unpow2 66.8%

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

      15. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in x around 0 64.0%

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

      1. associate-/r* 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

      4. unpow2 64.0%

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

      5. swap-sqr 77.6%

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

      6. unpow2 77.6%

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

      7. associate-/r* 77.6%

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

      8. unpow2 77.6%

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

      9. unpow2 77.6%

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

      10. swap-sqr 86.4%

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

      11. unpow2 86.4%

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

   7. Simplified 86.4%

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

      1. unpow2 86.4%

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

      2. associate-*l* 85.0%

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

      3. associate-*r* 82.8%

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

      4. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

       1. associate-*r* 84.2%

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

       2. associate-*r* 86.0%

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

       3. *-commutative 86.0%

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

       4. pow2 86.0%

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

       5. *-commutative 86.0%

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

       6. *-commutative 86.0%

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

       7. associate-*r* 87.7%

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

       8. *-un-lft-identity 87.7%

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

       9. pow-flip 87.7%

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

       10. metadata-eval 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. *-lft-identity 87.7%

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

       2. *-commutative 87.7%

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

       3. associate-*r* 86.4%

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

   13. Simplified 86.4%

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

       1. associate-*r* 87.7%

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

       2. *-commutative 87.7%

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

       3. metadata-eval 87.7%

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

       4. pow-prod-up 87.7%

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

       5. unpow-prod-down 87.7%

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

       6. *-commutative 87.7%

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

       7. associate-*r* 85.9%

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

       8. *-commutative 85.9%

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

       9. associate-*l* 85.1%

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

       10. inv-pow 85.1%

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

       11. associate-*l* 85.9%

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

       12. *-commutative 85.9%

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

       13. associate-*r* 87.7%

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

       14. *-commutative 87.7%

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

       15. associate-/r* 87.7%

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

       16. *-commutative 87.7%

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

       17. associate-*r* 85.9%

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

       18. associate-/r* 85.8%

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

       19. *-commutative 85.8%

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

       20. associate-*r* 86.4%

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

       21. *-commutative 86.4%

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

   15. Applied egg-rr 86.4%

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

   if 2.00000000000000005e-46 < x

   1. Initial program 75.6%

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

      1. unpow2 75.6%

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

   4. Applied egg-rr 75.6%

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

      1. *-un-lft-identity 75.6%

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

      2. pow2 75.6%

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

      3. times-frac 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 65.6%

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

      6. pow2 65.6%

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

      7. unpow-prod-down 80.8%

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

      8. times-frac 80.9%

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

      9. unpow-prod-down 97.1%

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

      10. associate-*l/ 97.0%

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

      11. add-sqr-sqrt 96.9%

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

      12. associate-*l* 96.9%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. div-inv 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r* 95.8%

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

      5. associate-/r* 95.8%

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

      6. associate-/r* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{x}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{s\_m}^{2} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{c\_m}}{c\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 5e+222)
   (/ (/ (cos (* x_m 2.0)) c_m) (* c_m (* s_m (* x_m (* x_m s_m)))))
   (/ (/ (/ 1.0 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (pow(s_m, 2.0) <= 5e+222) {
		tmp = (cos((x_m * 2.0)) / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
	} else {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_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 ((s_m ** 2.0d0) <= 5d+222) then
        tmp = (cos((x_m * 2.0d0)) / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))))
    else
        tmp = ((1.0d0 / (x_m * s_m)) / c_m) / ((x_m * s_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 (Math.pow(s_m, 2.0) <= 5e+222) {
		tmp = (Math.cos((x_m * 2.0)) / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
	} else {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_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 math.pow(s_m, 2.0) <= 5e+222:
		tmp = (math.cos((x_m * 2.0)) / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))))
	else:
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_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 ((s_m ^ 2.0) <= 5e+222)
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / c_m) / Float64(c_m * Float64(s_m * Float64(x_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_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 ((s_m ^ 2.0) <= 5e+222)
		tmp = (cos((x_m * 2.0)) / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
	else
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_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[N[Power[s$95$m, 2.0], $MachinePrecision], 5e+222], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * N[(s$95$m * N[(x$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 5.00000000000000023e222

   1. Initial program 74.8%

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

      1. associate-/r* 74.7%

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

      2. *-commutative 74.7%

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

      3. unpow2 74.7%

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

      4. sqr-neg 74.7%

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

      5. unpow2 74.7%

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

      6. cos-neg 74.7%

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

      7. *-commutative 74.7%

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

      8. distribute-rgt-neg-in 74.7%

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

      9. metadata-eval 74.7%

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

      10. unpow2 74.7%

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

      11. sqr-neg 74.7%

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

      12. unpow2 74.7%

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

      13. associate-*r* 67.1%

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

      14. unpow2 67.1%

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

      15. *-commutative 67.1%

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

   3. Simplified 67.1%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. unpow2 97.0%

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

      3. swap-sqr 79.9%

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

      4. swap-sqr 67.1%

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

      5. unpow2 67.1%

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

      6. associate-*r* 74.8%

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

      7. *-commutative 74.8%

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

      8. associate-*l* 78.8%

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

      9. *-un-lft-identity 78.8%

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

      10. associate-/r* 78.8%

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

      11. *-commutative 78.8%

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

      12. *-commutative 78.8%

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

      13. associate-*r* 70.3%

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

      14. pow2 70.3%

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

      15. unpow-prod-down 85.3%

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

   8. Applied egg-rr 85.3%

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

      1. unpow2 85.3%

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

      2. associate-*r* 81.4%

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

   10. Applied egg-rr 81.4%

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

   if 5.00000000000000023e222 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 69.2%

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

      1. associate-/r* 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. sqr-neg 69.2%

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

      5. unpow2 69.2%

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

      6. cos-neg 69.2%

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

      7. *-commutative 69.2%

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

      8. distribute-rgt-neg-in 69.2%

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

      9. metadata-eval 69.2%

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

      10. unpow2 69.2%

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

      11. sqr-neg 69.2%

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

      12. unpow2 69.2%

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

      13. associate-*r* 65.4%

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

      14. unpow2 65.4%

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

      15. *-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. associate-/r* 65.4%

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

      2. *-commutative 65.4%

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

      3. unpow2 65.4%

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

      4. unpow2 65.4%

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

      5. swap-sqr 88.4%

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

      6. unpow2 88.4%

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

      7. associate-/r* 88.4%

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

      8. unpow2 88.4%

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

      9. unpow2 88.4%

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

      10. swap-sqr 97.5%

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

      11. unpow2 97.5%

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

   7. Simplified 97.5%

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

      1. unpow2 97.5%

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

      2. associate-*l* 97.6%

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

      3. associate-*r* 93.0%

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

      4. *-commutative 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. associate-*r* 93.0%

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

       2. associate-*r* 93.1%

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

       3. *-commutative 93.1%

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

       4. pow2 93.1%

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

       5. *-commutative 93.1%

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

       6. *-commutative 93.1%

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

       7. associate-*r* 96.7%

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

       8. *-un-lft-identity 96.7%

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

       9. pow-flip 96.7%

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

       10. metadata-eval 96.7%

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

   11. Applied egg-rr 96.7%

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

       1. *-lft-identity 96.7%

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

       2. *-commutative 96.7%

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

       3. associate-*r* 97.5%

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

   13. Simplified 97.5%

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

       1. associate-*r* 96.7%

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

       2. *-commutative 96.7%

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

       3. metadata-eval 96.7%

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

       4. pow-prod-up 96.7%

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

       5. unpow-prod-down 96.7%

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

       6. *-commutative 96.7%

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

       7. associate-*r* 96.6%

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

       8. *-commutative 96.6%

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

       9. associate-*l* 96.5%

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

       10. inv-pow 96.5%

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

       11. associate-*l* 96.6%

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

       12. *-commutative 96.6%

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

       13. associate-*r* 96.7%

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

       14. *-commutative 96.7%

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

       15. associate-/r* 96.7%

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

       16. *-commutative 96.7%

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

       17. associate-*r* 96.5%

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

       18. associate-/r* 96.5%

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

       19. *-commutative 96.5%

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

       20. associate-*r* 97.6%

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

       21. *-commutative 97.6%

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

   15. Applied egg-rr 97.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{c \cdot \left(s \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 2.2e-46)
     (/ (/ (/ 1.0 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))
     (/ (/ (cos (* x_m 2.0)) t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 2.2e-46) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 2.2d-46) then
        tmp = ((1.0d0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 2.2e-46) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 2.2e-46:
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = (math.cos((x_m * 2.0)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 2.2e-46)
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 2.2e-46)
		tmp = ((1.0 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.2e-46], N[(N[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000001e-46

   1. Initial program 71.9%

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

      1. associate-/r* 71.9%

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

      2. *-commutative 71.9%

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

      3. unpow2 71.9%

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

      4. sqr-neg 71.9%

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

      5. unpow2 71.9%

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

      6. cos-neg 71.9%

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

      7. *-commutative 71.9%

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

      8. distribute-rgt-neg-in 71.9%

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

      9. metadata-eval 71.9%

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

      10. unpow2 71.9%

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

      11. sqr-neg 71.9%

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

      12. unpow2 71.9%

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

      13. associate-*r* 66.8%

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

      14. unpow2 66.8%

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

      15. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in x around 0 64.0%

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

      1. associate-/r* 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

      4. unpow2 64.0%

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

      5. swap-sqr 77.6%

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

      6. unpow2 77.6%

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

      7. associate-/r* 77.6%

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

      8. unpow2 77.6%

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

      9. unpow2 77.6%

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

      10. swap-sqr 86.4%

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

      11. unpow2 86.4%

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

   7. Simplified 86.4%

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

      1. unpow2 86.4%

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

      2. associate-*l* 85.0%

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

      3. associate-*r* 82.8%

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

      4. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

       1. associate-*r* 84.2%

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

       2. associate-*r* 86.0%

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

       3. *-commutative 86.0%

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

       4. pow2 86.0%

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

       5. *-commutative 86.0%

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

       6. *-commutative 86.0%

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

       7. associate-*r* 87.7%

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

       8. *-un-lft-identity 87.7%

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

       9. pow-flip 87.7%

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

       10. metadata-eval 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. *-lft-identity 87.7%

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

       2. *-commutative 87.7%

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

       3. associate-*r* 86.4%

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

   13. Simplified 86.4%

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

       1. associate-*r* 87.7%

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

       2. *-commutative 87.7%

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

       3. metadata-eval 87.7%

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

       4. pow-prod-up 87.7%

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

       5. unpow-prod-down 87.7%

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

       6. *-commutative 87.7%

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

       7. associate-*r* 85.9%

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

       8. *-commutative 85.9%

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

       9. associate-*l* 85.1%

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

       10. inv-pow 85.1%

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

       11. associate-*l* 85.9%

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

       12. *-commutative 85.9%

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

       13. associate-*r* 87.7%

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

       14. *-commutative 87.7%

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

       15. associate-/r* 87.7%

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

       16. *-commutative 87.7%

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

       17. associate-*r* 85.9%

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

       18. associate-/r* 85.8%

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

       19. *-commutative 85.8%

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

       20. associate-*r* 86.4%

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

       21. *-commutative 86.4%

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

   15. Applied egg-rr 86.4%

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

   if 2.2000000000000001e-46 < x

   1. Initial program 75.6%

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

      1. associate-/r* 75.5%

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

      2. *-commutative 75.5%

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

      3. unpow2 75.5%

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

      4. sqr-neg 75.5%

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

      5. unpow2 75.5%

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

      6. cos-neg 75.5%

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

      7. *-commutative 75.5%

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

      8. distribute-rgt-neg-in 75.5%

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

      9. metadata-eval 75.5%

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

      10. unpow2 75.5%

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

      11. sqr-neg 75.5%

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

      12. unpow2 75.5%

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

      13. associate-*r* 65.7%

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

      14. unpow2 65.7%

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

      15. *-commutative 65.7%

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

   3. Simplified 65.7%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.1%

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

      2. *-un-lft-identity 97.1%

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

      3. unpow2 97.1%

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

      4. associate-/r* 97.0%

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

      5. *-commutative 97.0%

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

      6. associate-*r* 95.8%

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

      7. *-commutative 95.8%

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

      8. associate-*r* 98.4%

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

      9. *-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. associate-/r* 72.9%

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

   2. *-commutative 72.9%

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

   3. unpow2 72.9%

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

   4. sqr-neg 72.9%

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

   5. unpow2 72.9%

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

   6. cos-neg 72.9%

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

   7. *-commutative 72.9%

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

   8. distribute-rgt-neg-in 72.9%

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

   9. metadata-eval 72.9%

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

   10. unpow2 72.9%

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

   11. sqr-neg 72.9%

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

   12. unpow2 72.9%

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

   13. associate-*r* 66.5%

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

   14. unpow2 66.5%

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

   15. *-commutative 66.5%

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

3. Simplified 66.5%

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

5. Taylor expanded in x around 0 62.9%

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

   1. associate-/r* 62.9%

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

   2. *-commutative 62.9%

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

   3. unpow2 62.9%

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

   4. unpow2 62.9%

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

   5. swap-sqr 73.7%

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

   6. unpow2 73.7%

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

   7. associate-/r* 73.7%

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

   8. unpow2 73.7%

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

   9. unpow2 73.7%

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

   10. swap-sqr 82.9%

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

   11. unpow2 82.9%

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

7. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*l* 81.6%

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

   3. associate-*r* 79.9%

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

   4. *-commutative 79.9%

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

9. Applied egg-rr 79.9%

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

    1. associate-*r* 81.3%

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

    2. associate-*r* 82.6%

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

    3. *-commutative 82.6%

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

    4. pow2 82.6%

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

    5. *-commutative 82.6%

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

    6. *-commutative 82.6%

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

    7. associate-*r* 83.9%

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

    8. *-un-lft-identity 83.9%

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

    9. pow-flip 83.9%

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

    10. metadata-eval 83.9%

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

11. Applied egg-rr 83.9%

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

    1. *-lft-identity 83.9%

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

    2. *-commutative 83.9%

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

    3. associate-*r* 83.0%

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

13. Simplified 83.0%

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

    1. associate-*r* 83.9%

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

    2. *-commutative 83.9%

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

    3. metadata-eval 83.9%

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

    4. pow-prod-up 83.9%

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

    5. unpow-prod-down 83.9%

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

    6. *-commutative 83.9%

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

    7. associate-*r* 82.5%

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

    8. *-commutative 82.5%

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

    9. associate-*l* 81.4%

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

    10. inv-pow 81.4%

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

    11. associate-*l* 82.5%

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

    12. *-commutative 82.5%

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

    13. associate-*r* 83.9%

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

    14. *-commutative 83.9%

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

    15. associate-/r* 83.9%

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

    16. *-commutative 83.9%

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

    17. associate-*r* 82.5%

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

    18. associate-/r* 82.5%

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

    19. *-commutative 82.5%

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

    20. associate-*r* 82.9%

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

    21. *-commutative 82.9%

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

15. Applied egg-rr 82.9%

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

16. Final simplification 82.9%

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

Alternative 5: 78.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. associate-/r* 72.9%

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

   2. *-commutative 72.9%

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

   3. unpow2 72.9%

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

   4. sqr-neg 72.9%

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

   5. unpow2 72.9%

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

   6. cos-neg 72.9%

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

   7. *-commutative 72.9%

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

   8. distribute-rgt-neg-in 72.9%

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

   9. metadata-eval 72.9%

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

   10. unpow2 72.9%

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

   11. sqr-neg 72.9%

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

   12. unpow2 72.9%

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

   13. associate-*r* 66.5%

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

   14. unpow2 66.5%

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

   15. *-commutative 66.5%

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

3. Simplified 66.5%

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

5. Taylor expanded in x around 0 62.9%

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

   1. associate-/r* 62.9%

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

   2. *-commutative 62.9%

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

   3. unpow2 62.9%

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

   4. unpow2 62.9%

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

   5. swap-sqr 73.7%

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

   6. unpow2 73.7%

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

   7. associate-/r* 73.7%

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

   8. unpow2 73.7%

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

   9. unpow2 73.7%

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

   10. swap-sqr 82.9%

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

   11. unpow2 82.9%

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

7. Simplified 82.9%

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

8. Taylor expanded in c around 0 62.9%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

   3. associate-*r* 62.8%

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

   4. unpow2 62.8%

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

   5. unpow2 62.8%

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

   6. unpow2 62.8%

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

   7. swap-sqr 71.9%

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

   8. swap-sqr 82.6%

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

   9. unpow2 82.6%

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

   10. *-commutative 82.6%

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

   11. *-commutative 82.6%

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

   12. associate-*l* 83.9%

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

10. Simplified 83.9%

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

    1. unpow2 83.9%

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

12. Applied egg-rr 83.9%

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

13. Final simplification 83.9%

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

Alternative 6: 79.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. associate-/r* 72.9%

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

   2. *-commutative 72.9%

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

   3. unpow2 72.9%

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

   4. sqr-neg 72.9%

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

   5. unpow2 72.9%

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

   6. cos-neg 72.9%

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

   7. *-commutative 72.9%

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

   8. distribute-rgt-neg-in 72.9%

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

   9. metadata-eval 72.9%

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

   10. unpow2 72.9%

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

   11. sqr-neg 72.9%

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

   12. unpow2 72.9%

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

   13. associate-*r* 66.5%

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

   14. unpow2 66.5%

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

   15. *-commutative 66.5%

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

3. Simplified 66.5%

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

5. Taylor expanded in x around 0 62.9%

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

   1. associate-/r* 62.9%

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

   2. *-commutative 62.9%

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

   3. unpow2 62.9%

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

   4. unpow2 62.9%

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

   5. swap-sqr 73.7%

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

   6. unpow2 73.7%

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

   7. associate-/r* 73.7%

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

   8. unpow2 73.7%

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

   9. unpow2 73.7%

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

   10. swap-sqr 82.9%

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

   11. unpow2 82.9%

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

7. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*l* 81.6%

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

   3. associate-*r* 79.9%

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

   4. *-commutative 79.9%

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

9. Applied egg-rr 79.9%

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

10. Taylor expanded in s around 0 81.6%

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

11. Final simplification 81.6%

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

Reproduce

herbie shell --seed 2024169 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))