mixedcos

Percentage Accurate: 66.0% → 99.2%

Time: 15.0s

Alternatives: 8

Speedup: 24.1×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 7 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{s\_m}}{\left(c\_m \cdot x\_m\right) \cdot \left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 7e+27)
     (* (/ 1.0 t_0) (/ t_1 t_0))
     (/ (/ t_1 s_m) (* (* c_m x_m) (* (* c_m x_m) s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 7e+27) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = (t_1 / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 7d+27) then
        tmp = (1.0d0 / t_0) * (t_1 / t_0)
    else
        tmp = (t_1 / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 7e+27) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = (t_1 / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 7e+27:
		tmp = (1.0 / t_0) * (t_1 / t_0)
	else:
		tmp = (t_1 / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 7e+27)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / t_0));
	else
		tmp = Float64(Float64(t_1 / s_m) / Float64(Float64(c_m * x_m) * Float64(Float64(c_m * x_m) * s_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 7e+27)
		tmp = (1.0 / t_0) * (t_1 / t_0);
	else
		tmp = (t_1 / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 7e+27], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / s$95$m), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.0000000000000004e27

   1. Initial program 71.2%

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

      1. associate-/r* 71.1%

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

      2. *-commutative 71.1%

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

      3. unpow2 71.1%

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

      4. sqr-neg 71.1%

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

      5. unpow2 71.1%

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

      6. cos-neg 71.1%

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

      7. *-commutative 71.1%

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

      8. distribute-rgt-neg-in 71.1%

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

      9. metadata-eval 71.1%

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

      10. unpow2 71.1%

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

      11. sqr-neg 71.1%

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

      12. unpow2 71.1%

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

      13. associate-*r* 64.4%

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

      14. unpow2 64.4%

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

      15. *-commutative 64.4%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 99.1%

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

   if 7.0000000000000004e27 < x

   1. Initial program 69.5%

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

      1. associate-/r* 69.5%

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

      2. *-commutative 69.5%

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

      3. unpow2 69.5%

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

      4. sqr-neg 69.5%

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

      5. unpow2 69.5%

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

      6. cos-neg 69.5%

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

      7. *-commutative 69.5%

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

      8. distribute-rgt-neg-in 69.5%

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

      9. metadata-eval 69.5%

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

      10. unpow2 69.5%

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

      11. sqr-neg 69.5%

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

      12. unpow2 69.5%

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

      13. associate-*r* 52.8%

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

      14. unpow2 52.8%

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

      15. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   6. Applied egg-rr 94.9%

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

      1. *-un-lft-identity 94.9%

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

      2. associate-*r* 91.9%

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

      3. times-frac 91.8%

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

      4. *-commutative 91.8%

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

   8. Applied egg-rr 91.8%

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

      1. associate-*l/ 91.9%

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

      2. *-lft-identity 91.9%

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

   10. Simplified 91.9%

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

       1. clear-num 91.9%

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

       2. frac-times 90.3%

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

       3. *-un-lft-identity 90.3%

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

       4. *-commutative 90.3%

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

       5. /-rgt-identity 90.3%

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

       6. *-commutative 90.3%

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

       7. associate-*r* 95.0%

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

       8. *-commutative 95.0%

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

   12. Applied egg-rr 95.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(c \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{s\_m}}{\left(c\_m \cdot x\_m\right) \cdot \left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.5e-8)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (/ (cos (* x_m 2.0)) s_m) (* (* c_m x_m) (* (* c_m x_m) s_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.5e-8) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (cos((x_m * 2.0)) / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	}
	return tmp;
}

Fortran

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

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.5e-8) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.5e-8:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (math.cos((x_m * 2.0)) / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.5e-8)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / s_m) / Float64(Float64(c_m * x_m) * Float64(Float64(c_m * x_m) * s_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.5e-8)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (cos((x_m * 2.0)) / s_m) / ((c_m * x_m) * ((c_m * x_m) * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.5e-8], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999987e-8

   1. Initial program 70.3%

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

      1. associate-/r* 70.2%

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

      2. *-commutative 70.2%

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

      3. unpow2 70.2%

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

      4. sqr-neg 70.2%

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

      5. unpow2 70.2%

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

      6. cos-neg 70.2%

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

      7. *-commutative 70.2%

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

      8. distribute-rgt-neg-in 70.2%

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

      9. metadata-eval 70.2%

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

      10. unpow2 70.2%

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

      11. sqr-neg 70.2%

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

      12. unpow2 70.2%

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

      13. associate-*r* 63.3%

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

      14. unpow2 63.3%

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

      15. *-commutative 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. associate-/r* 58.3%

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

      2. *-commutative 58.3%

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

      3. unpow2 58.3%

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

      4. unpow2 58.3%

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

      5. swap-sqr 70.3%

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

      6. unpow2 70.3%

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

      7. associate-/r* 70.1%

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

      8. unpow2 70.1%

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

      9. unpow2 70.1%

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

      10. swap-sqr 86.8%

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

      11. unpow2 86.8%

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

   7. Simplified 86.8%

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

   8. Taylor expanded in c around 0 58.3%

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

      1. associate-*r* 57.8%

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

      2. unpow2 57.8%

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

      3. unpow2 57.8%

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

      4. swap-sqr 68.1%

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

      5. unpow2 68.1%

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

      6. swap-sqr 84.3%

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

      7. associate-/l/ 84.3%

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

      8. *-rgt-identity 84.3%

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

      9. associate-*r/ 84.2%

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

      10. unpow-1 84.2%

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

      11. unpow-1 84.2%

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

      12. pow-sqr 84.3%

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

      13. associate-*r* 87.1%

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

      14. metadata-eval 87.1%

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

   10. Simplified 87.1%

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

   if 1.49999999999999987e-8 < x

   1. Initial program 72.3%

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

      1. associate-/r* 72.3%

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

      2. *-commutative 72.3%

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

      3. unpow2 72.3%

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

      4. sqr-neg 72.3%

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

      5. unpow2 72.3%

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

      6. cos-neg 72.3%

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

      7. *-commutative 72.3%

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

      8. distribute-rgt-neg-in 72.3%

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

      9. metadata-eval 72.3%

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

      10. unpow2 72.3%

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

      11. sqr-neg 72.3%

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

      12. unpow2 72.3%

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

      13. associate-*r* 57.1%

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

      14. unpow2 57.1%

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

      15. *-commutative 57.1%

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

   3. Simplified 57.1%

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

   6. Applied egg-rr 95.3%

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

      1. *-un-lft-identity 95.3%

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

      2. associate-*r* 92.6%

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

      3. times-frac 92.6%

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

      4. *-commutative 92.6%

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

   8. Applied egg-rr 92.6%

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

      1. associate-*l/ 92.6%

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

      2. *-lft-identity 92.6%

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

   10. Simplified 92.6%

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

       1. clear-num 92.6%

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

       2. frac-times 91.2%

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

       3. *-un-lft-identity 91.2%

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

       4. *-commutative 91.2%

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

       5. /-rgt-identity 91.2%

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

       6. *-commutative 91.2%

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

       7. associate-*r* 95.4%

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

       8. *-commutative 95.4%

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

   12. Applied egg-rr 95.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(c \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 2.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (* (/ (/ 1.0 (* c_m x_m)) s_m) (/ (/ (cos (* x_m 2.0)) s_m) (* c_m x_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) * ((cos((x_m * 2.0)) / s_m) / (c_m * x_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) * ((cos((x_m * 2.0d0)) / s_m) / (c_m * x_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) * ((Math.cos((x_m * 2.0)) / s_m) / (c_m * x_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) * ((math.cos((x_m * 2.0)) / s_m) / (c_m * x_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(c_m * x_m)) / s_m) * Float64(Float64(cos(Float64(x_m * 2.0)) / s_m) / Float64(c_m * x_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) * ((cos((x_m * 2.0)) / s_m) / (c_m * x_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[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision] * N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

6. Applied egg-rr 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. associate-*r* 96.4%

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

   3. times-frac 96.4%

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

   4. *-commutative 96.4%

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

8. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-lft-identity 96.4%

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

10. Simplified 96.4%

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

    1. inv-pow 96.4%

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

    2. associate-*r* 97.8%

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

    3. unpow-prod-down 97.8%

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

12. Applied egg-rr 97.8%

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

    1. unpow-1 97.8%

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

    2. *-commutative 97.8%

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

    3. associate-/r* 97.8%

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

    4. unpow-1 97.8%

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

    5. associate-*l/ 97.9%

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

    6. associate-/r* 97.8%

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

    7. *-lft-identity 97.8%

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

14. Simplified 97.8%

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

Alternative 4: 91.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

6. Applied egg-rr 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. associate-*r* 96.4%

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

   3. times-frac 96.4%

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

   4. *-commutative 96.4%

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

8. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-lft-identity 96.4%

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

10. Simplified 96.4%

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

    1. *-commutative 96.4%

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

    2. associate-/r* 96.4%

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

    3. associate-/l/ 96.5%

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

    4. associate-*r* 98.1%

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

    5. *-commutative 98.1%

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

    6. frac-times 94.3%

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

    7. div-inv 94.3%

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

    8. *-commutative 94.3%

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

    9. *-commutative 94.3%

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

    10. associate-*r* 92.5%

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

    11. *-commutative 92.5%

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

12. Applied egg-rr 92.5%

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

13. Final simplification 92.5%

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

Alternative 5: 79.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

5. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. *-commutative 55.5%

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

   3. unpow2 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 66.5%

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

   6. unpow2 66.5%

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

   7. associate-/r* 66.4%

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

   8. unpow2 66.4%

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

   9. unpow2 66.4%

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

   10. swap-sqr 79.9%

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

   11. unpow2 79.9%

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

7. Simplified 79.9%

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

8. Taylor expanded in c around 0 55.6%

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

   1. associate-*r* 55.1%

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

   2. unpow2 55.1%

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

   3. unpow2 55.1%

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

   4. swap-sqr 64.6%

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

   5. unpow2 64.6%

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

   6. swap-sqr 77.9%

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

   7. associate-/l/ 77.9%

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

   8. *-rgt-identity 77.9%

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

   9. associate-*r/ 77.9%

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

   10. unpow-1 77.9%

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

   11. unpow-1 77.9%

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

   12. pow-sqr 77.9%

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

   13. associate-*r* 80.1%

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

   14. metadata-eval 80.1%

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

10. Simplified 80.1%

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

11. Final simplification 80.1%

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

Alternative 6: 79.0% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in x around 0 80.1%

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

Alternative 7: 76.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

5. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. *-commutative 55.5%

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

   3. unpow2 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 66.5%

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

   6. unpow2 66.5%

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

   7. associate-/r* 66.4%

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

   8. unpow2 66.4%

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

   9. unpow2 66.4%

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

   10. swap-sqr 79.9%

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

   11. unpow2 79.9%

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

7. Simplified 79.9%

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

8. Taylor expanded in c around 0 55.6%

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

   1. associate-*r* 55.1%

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

   2. unpow2 55.1%

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

   3. unpow2 55.1%

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

   4. swap-sqr 64.6%

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

   5. unpow2 64.6%

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

   6. swap-sqr 77.9%

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

   7. associate-/l/ 77.9%

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

   8. *-rgt-identity 77.9%

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

   9. associate-*r/ 77.9%

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

   10. unpow-1 77.9%

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

   11. unpow-1 77.9%

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

   12. pow-sqr 77.9%

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

   13. associate-*r* 80.1%

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

   14. metadata-eval 80.1%

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

10. Simplified 80.1%

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

    1. metadata-eval 80.1%

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

    2. pow-prod-up 80.1%

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

    3. *-commutative 80.1%

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

    4. associate-*r* 79.4%

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

    5. unpow-prod-down 79.4%

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

    6. inv-pow 79.4%

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

    7. inv-pow 79.4%

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

    8. associate-/l/ 79.4%

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

    9. div-inv 79.4%

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

    10. inv-pow 79.4%

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

    11. div-inv 79.4%

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

    12. *-commutative 79.4%

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

    13. associate-*r* 79.5%

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

    14. *-commutative 79.5%

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

    15. div-inv 79.5%

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

    16. associate-/l/ 79.4%

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

    17. frac-times 79.4%

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

    18. metadata-eval 79.4%

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

    19. *-commutative 79.4%

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

12. Applied egg-rr 79.4%

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

13. Final simplification 79.4%

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

Alternative 8: 78.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. associate-/r* 70.8%

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

   2. *-commutative 70.8%

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

   3. unpow2 70.8%

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

   4. sqr-neg 70.8%

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

   5. unpow2 70.8%

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

   6. cos-neg 70.8%

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

   7. *-commutative 70.8%

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

   8. distribute-rgt-neg-in 70.8%

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

   9. metadata-eval 70.8%

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

   10. unpow2 70.8%

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

   11. sqr-neg 70.8%

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

   12. unpow2 70.8%

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

   13. associate-*r* 61.7%

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

   14. unpow2 61.7%

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

   15. *-commutative 61.7%

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

3. Simplified 61.7%

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

5. Taylor expanded in x around 0 55.6%

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

   1. associate-/r* 55.5%

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

   2. *-commutative 55.5%

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

   3. unpow2 55.5%

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

   4. unpow2 55.5%

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

   5. swap-sqr 66.5%

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

   6. unpow2 66.5%

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

   7. associate-/r* 66.4%

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

   8. unpow2 66.4%

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

   9. unpow2 66.4%

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

   10. swap-sqr 79.9%

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

   11. unpow2 79.9%

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

7. Simplified 79.9%

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

   1. /-rgt-identity 79.9%

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

   2. clear-num 79.9%

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

   3. pow-flip 79.9%

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

   4. *-commutative 79.9%

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

   5. metadata-eval 79.9%

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

   6. unpow-prod-down 67.6%

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

   7. metadata-eval 67.6%

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

   8. pow-flip 66.3%

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

   9. div-inv 66.3%

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

   10. clear-num 66.3%

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

   11. unpow2 66.3%

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

   12. associate-/l* 73.5%

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

9. Applied egg-rr 73.5%

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

    1. *-un-lft-identity 73.5%

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

    2. metadata-eval 73.5%

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

    3. pow-sqr 73.5%

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

    4. inv-pow 73.5%

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

    5. inv-pow 73.5%

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

    6. times-frac 77.2%

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

    7. remove-double-div 77.2%

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

    8. associate-/r/ 77.2%

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

    9. /-rgt-identity 77.2%

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

    10. *-commutative 77.2%

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

    11. associate-*l* 75.2%

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

    12. *-commutative 75.2%

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

    13. associate-*r* 76.6%

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

    14. *-commutative 76.6%

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

11. Applied egg-rr 76.6%

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

12. Taylor expanded in s around 0 77.2%

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

    1. *-commutative 77.2%

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

14. Simplified 77.2%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))