mixedcos

Percentage Accurate: 67.5% → 99.3%

Time: 14.5s

Alternatives: 10

Speedup: 24.1×

• 33.0% 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: 67.5% 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: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 1.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot s\_m}}{c\_m} \cdot \frac{t\_0}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{s\_m}}{x\_m \cdot c\_m}}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))))
   (if (<= x_m 1.3e-107)
     (* (/ (/ 1.0 (* x_m s_m)) c_m) (/ t_0 (* (* x_m s_m) c_m)))
     (/ (/ (/ t_0 s_m) (* x_m c_m)) (* s_m (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.3e-107) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) * (t_0 / ((x_m * s_m) * c_m));
	} else {
		tmp = ((t_0 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    if (x_m <= 1.3d-107) then
        tmp = ((1.0d0 / (x_m * s_m)) / c_m) * (t_0 / ((x_m * s_m) * c_m))
    else
        tmp = ((t_0 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.3e-107) {
		tmp = ((1.0 / (x_m * s_m)) / c_m) * (t_0 / ((x_m * s_m) * c_m));
	} else {
		tmp = ((t_0 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 1.3e-107:
		tmp = ((1.0 / (x_m * s_m)) / c_m) * (t_0 / ((x_m * s_m) * c_m))
	else:
		tmp = ((t_0 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 1.3e-107)
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * s_m)) / c_m) * Float64(t_0 / Float64(Float64(x_m * s_m) * c_m)));
	else
		tmp = Float64(Float64(Float64(t_0 / s_m) / Float64(x_m * c_m)) / Float64(s_m * Float64(x_m * c_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 1.3e-107)
		tmp = ((1.0 / (x_m * s_m)) / c_m) * (t_0 / ((x_m * s_m) * c_m));
	else
		tmp = ((t_0 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 1.3e-107], N[(N[(N[(1.0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(t$95$0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3e-107

   1. Initial program 65.5%

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

      1. unpow2 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Applied egg-rr 97.5%

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

   if 1.3e-107 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. unpow2 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. div-inv 72.3%

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

      2. *-commutative 72.3%

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

      3. frac-2neg 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*r* 65.6%

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

      7. pow2 65.6%

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

      8. unpow-prod-down 79.5%

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

      9. distribute-rgt-neg-in 79.5%

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

      10. pow2 79.5%

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

      11. *-commutative 79.5%

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

      12. unpow-prod-down 98.6%

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

      13. frac-2neg 98.6%

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

      14. pow-flip 98.6%

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

      15. *-commutative 98.6%

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

      16. metadata-eval 98.6%

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

      17. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-div 98.6%

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

      4. inv-pow 98.6%

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

      5. pow1 98.6%

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

      6. associate-*r/ 98.5%

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

      7. *-commutative 98.5%

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

      8. div-inv 98.6%

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

      9. *-commutative 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-*l* 96.3%

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

      12. *-commutative 96.3%

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

      13. associate-*l* 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in x around inf 96.2%

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

       1. *-commutative 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r* 97.2%

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

       4. associate-/r* 97.2%

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

       5. *-commutative 97.2%

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

   11. Simplified 97.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{c} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(x\_m \cdot s\_m\right) \cdot c\_m\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;s\_m \leq 2.4 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{s\_m}}{x\_m \cdot c\_m}}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0} \cdot \frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* x_m s_m) c_m)) (t_1 (cos (* x_m 2.0))))
   (if (<= s_m 2.4e+217)
     (/ (/ (/ t_1 s_m) (* x_m c_m)) (* s_m (* x_m c_m)))
     (* (/ t_1 t_0) (/ 1.0 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;
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (s_m <= 2.4e+217) {
		tmp = ((t_1 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	} else {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m * s_m) * c_m
    t_1 = cos((x_m * 2.0d0))
    if (s_m <= 2.4d+217) then
        tmp = ((t_1 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
    else
        tmp = (t_1 / t_0) * (1.0d0 / 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 = (x_m * s_m) * c_m;
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (s_m <= 2.4e+217) {
		tmp = ((t_1 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	} else {
		tmp = (t_1 / t_0) * (1.0 / 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 = (x_m * s_m) * c_m
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if s_m <= 2.4e+217:
		tmp = ((t_1 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
	else:
		tmp = (t_1 / t_0) * (1.0 / 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(Float64(x_m * s_m) * c_m)
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (s_m <= 2.4e+217)
		tmp = Float64(Float64(Float64(t_1 / s_m) / Float64(x_m * c_m)) / Float64(s_m * Float64(x_m * c_m)));
	else
		tmp = Float64(Float64(t_1 / t_0) * Float64(1.0 / 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 = (x_m * s_m) * c_m;
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (s_m <= 2.4e+217)
		tmp = ((t_1 / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	else
		tmp = (t_1 / t_0) * (1.0 / 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[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s$95$m, 2.4e+217], N[(N[(N[(t$95$1 / s$95$m), $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < 2.3999999999999998e217

   1. Initial program 68.1%

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

      1. unpow2 68.1%

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

   4. Applied egg-rr 68.1%

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

      1. div-inv 68.1%

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

      2. *-commutative 68.1%

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

      3. frac-2neg 68.1%

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

      4. distribute-rgt-neg-in 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*r* 61.1%

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

      7. pow2 61.1%

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

      8. unpow-prod-down 78.5%

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

      9. distribute-rgt-neg-in 78.5%

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

      10. pow2 78.5%

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

      11. *-commutative 78.5%

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

      12. unpow-prod-down 97.4%

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

      13. frac-2neg 97.4%

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

      14. pow-flip 97.7%

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

      15. *-commutative 97.7%

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

      16. metadata-eval 97.7%

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

      17. *-commutative 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. metadata-eval 97.7%

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

      3. pow-div 97.7%

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

      4. inv-pow 97.7%

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

      5. pow1 97.7%

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

      6. associate-*r/ 97.7%

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

      7. *-commutative 97.7%

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

      8. div-inv 97.7%

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

      9. *-commutative 97.7%

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

      10. *-commutative 97.7%

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

      11. associate-*l* 94.8%

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

      12. *-commutative 94.8%

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

      13. associate-*l* 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in x around inf 94.8%

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

       1. *-commutative 94.8%

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

       2. *-commutative 94.8%

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

       3. associate-*r* 96.3%

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

       4. associate-/r* 96.3%

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

       5. *-commutative 96.3%

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

   11. Simplified 96.3%

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

   if 2.3999999999999998e217 < s

   1. Initial program 61.1%

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

      1. *-un-lft-identity 61.1%

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

      2. add-sqr-sqrt 61.1%

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

      3. times-frac 61.1%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.4 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot s\right) \cdot c} \cdot \frac{1}{\left(x \cdot s\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{if}\;x\_m \leq 1.26 \cdot 10^{-107}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x\_m \cdot 2\right)}{s\_m}}{x\_m \cdot c\_m}}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (* x_m s_m) c_m))))
   (if (<= x_m 1.26e-107)
     (* t_0 t_0)
     (/ (/ (/ (cos (* x_m 2.0)) s_m) (* x_m c_m)) (* s_m (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / ((x_m * s_m) * c_m);
	double tmp;
	if (x_m <= 1.26e-107) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((cos((x_m * 2.0)) / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / ((x_m * s_m) * c_m)
    if (x_m <= 1.26d-107) then
        tmp = t_0 * t_0
    else
        tmp = ((cos((x_m * 2.0d0)) / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / ((x_m * s_m) * c_m);
	double tmp;
	if (x_m <= 1.26e-107) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((Math.cos((x_m * 2.0)) / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = 1.0 / ((x_m * s_m) * c_m)
	tmp = 0
	if x_m <= 1.26e-107:
		tmp = t_0 * t_0
	else:
		tmp = ((math.cos((x_m * 2.0)) / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(1.0 / Float64(Float64(x_m * s_m) * c_m))
	tmp = 0.0
	if (x_m <= 1.26e-107)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(Float64(Float64(cos(Float64(x_m * 2.0)) / s_m) / Float64(x_m * c_m)) / Float64(s_m * Float64(x_m * c_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = 1.0 / ((x_m * s_m) * c_m);
	tmp = 0.0;
	if (x_m <= 1.26e-107)
		tmp = t_0 * t_0;
	else
		tmp = ((cos((x_m * 2.0)) / s_m) / (x_m * c_m)) / (s_m * (x_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.26e-107], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2599999999999999e-107

   1. Initial program 65.5%

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

      1. *-un-lft-identity 65.5%

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

      2. add-sqr-sqrt 65.5%

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

      3. times-frac 65.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in x around 0 85.6%

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

   if 1.2599999999999999e-107 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. unpow2 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. div-inv 72.3%

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

      2. *-commutative 72.3%

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

      3. frac-2neg 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*r* 65.6%

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

      7. pow2 65.6%

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

      8. unpow-prod-down 79.5%

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

      9. distribute-rgt-neg-in 79.5%

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

      10. pow2 79.5%

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

      11. *-commutative 79.5%

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

      12. unpow-prod-down 98.6%

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

      13. frac-2neg 98.6%

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

      14. pow-flip 98.6%

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

      15. *-commutative 98.6%

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

      16. metadata-eval 98.6%

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

      17. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-div 98.6%

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

      4. inv-pow 98.6%

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

      5. pow1 98.6%

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

      6. associate-*r/ 98.5%

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

      7. *-commutative 98.5%

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

      8. div-inv 98.6%

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

      9. *-commutative 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-*l* 96.3%

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

      12. *-commutative 96.3%

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

      13. associate-*l* 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in x around inf 96.2%

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

       1. *-commutative 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r* 97.2%

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

       4. associate-/r* 97.2%

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

       5. *-commutative 97.2%

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

   11. Simplified 97.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot c} \cdot \frac{1}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \frac{1}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{if}\;x\_m \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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))) (t_1 (/ 1.0 (* (* x_m s_m) c_m))))
   (if (<= x_m 1.16e-107) (* t_1 t_1) (/ (/ (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 t_1 = 1.0 / ((x_m * s_m) * c_m);
	double tmp;
	if (x_m <= 1.16e-107) {
		tmp = t_1 * t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = 1.0d0 / ((x_m * s_m) * c_m)
    if (x_m <= 1.16d-107) then
        tmp = t_1 * t_1
    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 t_1 = 1.0 / ((x_m * s_m) * c_m);
	double tmp;
	if (x_m <= 1.16e-107) {
		tmp = t_1 * t_1;
	} 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)
	t_1 = 1.0 / ((x_m * s_m) * c_m)
	tmp = 0
	if x_m <= 1.16e-107:
		tmp = t_1 * t_1
	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))
	t_1 = Float64(1.0 / Float64(Float64(x_m * s_m) * c_m))
	tmp = 0.0
	if (x_m <= 1.16e-107)
		tmp = Float64(t_1 * t_1);
	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);
	t_1 = 1.0 / ((x_m * s_m) * c_m);
	tmp = 0.0;
	if (x_m <= 1.16e-107)
		tmp = t_1 * t_1;
	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]}, Block[{t$95$1 = N[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.16e-107], N[(t$95$1 * t$95$1), $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 < 1.16e-107

   1. Initial program 65.5%

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

      1. *-un-lft-identity 65.5%

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

      2. add-sqr-sqrt 65.5%

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

      3. times-frac 65.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in x around 0 85.6%

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

   if 1.16e-107 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. unpow2 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. div-inv 72.3%

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

      2. *-commutative 72.3%

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

      3. frac-2neg 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*r* 65.6%

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

      7. pow2 65.6%

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

      8. unpow-prod-down 79.5%

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

      9. distribute-rgt-neg-in 79.5%

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

      10. pow2 79.5%

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

      11. *-commutative 79.5%

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

      12. unpow-prod-down 98.6%

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

      13. frac-2neg 98.6%

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

      14. pow-flip 98.6%

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

      15. *-commutative 98.6%

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

      16. metadata-eval 98.6%

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

      17. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-div 98.6%

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

      4. inv-pow 98.6%

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

      5. pow1 98.6%

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

      6. associate-*r/ 98.5%

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

      7. *-commutative 98.5%

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

      8. div-inv 98.6%

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

      9. *-commutative 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-*l* 96.3%

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

      12. *-commutative 96.3%

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

      13. associate-*l* 97.2%

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

   8. Applied egg-rr 97.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot c} \cdot \frac{1}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(x\_m \cdot s\_m\right) \cdot c\_m\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 9.2 \cdot 10^{-108}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* x_m s_m) c_m)) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 9.2e-108)
     (* t_1 t_1)
     (/ (/ (cos (* x_m 2.0)) t_0) (* s_m (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 9.2e-108) {
		tmp = t_1 * t_1;
	} else {
		tmp = (cos((x_m * 2.0)) / t_0) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m * s_m) * c_m
    t_1 = 1.0d0 / t_0
    if (x_m <= 9.2d-108) then
        tmp = t_1 * t_1
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) / (s_m * (x_m * c_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 9.2e-108) {
		tmp = t_1 * t_1;
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_0) / (s_m * (x_m * c_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (x_m * s_m) * c_m
	t_1 = 1.0 / t_0
	tmp = 0
	if x_m <= 9.2e-108:
		tmp = t_1 * t_1
	else:
		tmp = (math.cos((x_m * 2.0)) / t_0) / (s_m * (x_m * c_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(x_m * s_m) * c_m)
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 9.2e-108)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / Float64(s_m * Float64(x_m * c_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (x_m * s_m) * c_m;
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x_m <= 9.2e-108)
		tmp = t_1 * t_1;
	else
		tmp = (cos((x_m * 2.0)) / t_0) / (s_m * (x_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 9.2e-108], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.19999999999999983e-108

   1. Initial program 65.5%

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

      1. *-un-lft-identity 65.5%

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

      2. add-sqr-sqrt 65.5%

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

      3. times-frac 65.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in x around 0 85.6%

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

   if 9.19999999999999983e-108 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. unpow2 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. div-inv 72.3%

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

      2. *-commutative 72.3%

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

      3. frac-2neg 72.3%

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

      4. distribute-rgt-neg-in 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*r* 65.6%

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

      7. pow2 65.6%

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

      8. unpow-prod-down 79.5%

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

      9. distribute-rgt-neg-in 79.5%

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

      10. pow2 79.5%

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

      11. *-commutative 79.5%

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

      12. unpow-prod-down 98.6%

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

      13. frac-2neg 98.6%

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

      14. pow-flip 98.6%

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

      15. *-commutative 98.6%

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

      16. metadata-eval 98.6%

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

      17. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-div 98.6%

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

      4. inv-pow 98.6%

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

      5. pow1 98.6%

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

      6. associate-*r/ 98.5%

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

      7. *-commutative 98.5%

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

      8. div-inv 98.6%

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

      9. *-commutative 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-*l* 96.3%

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

      12. *-commutative 96.3%

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

      13. associate-*l* 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in s around 0 96.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot c} \cdot \frac{1}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot s\right) \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

4. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 97.5%

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

   2. frac-2neg 97.5%

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

   3. add-sqr-sqrt 0.0%

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

   4. associate-/r* 0.0%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 7: 80.0% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in x around 0 55.0%

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

   1. associate-/r* 55.0%

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

   2. *-commutative 55.0%

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

   3. unpow2 55.0%

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

   4. unpow2 55.0%

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

   5. swap-sqr 68.2%

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

   6. unpow2 68.2%

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

   7. associate-/r* 68.9%

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

   8. unpow2 68.9%

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

   9. unpow2 68.9%

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

   10. swap-sqr 79.7%

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

   11. unpow2 79.7%

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

   12. *-commutative 79.7%

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

5. Simplified 79.7%

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

   1. metadata-eval 79.7%

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

   2. *-commutative 79.7%

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

   3. unpow2 79.7%

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

   4. frac-times 79.8%

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

   5. *-commutative 79.8%

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

   6. associate-/r* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-/r* 79.8%

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

7. Applied egg-rr 79.8%

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

Alternative 8: 80.0% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. *-un-lft-identity 67.7%

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

   2. add-sqr-sqrt 67.6%

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

   3. times-frac 67.6%

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

4. Applied egg-rr 97.8%

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

5. Taylor expanded in x around 0 79.8%

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

6. Final simplification 79.8%

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

Alternative 9: 80.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in x around 0 55.0%

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

   1. associate-/r* 55.0%

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

   2. *-commutative 55.0%

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

   3. unpow2 55.0%

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

   4. unpow2 55.0%

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

   5. swap-sqr 68.2%

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

   6. unpow2 68.2%

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

   7. associate-/r* 68.9%

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

   8. unpow2 68.9%

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

   9. unpow2 68.9%

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

   10. swap-sqr 79.7%

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

   11. unpow2 79.7%

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

   12. *-commutative 79.7%

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

5. Simplified 79.7%

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

   1. *-commutative 79.7%

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

   2. unpow2 79.7%

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

7. Applied egg-rr 79.7%

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

8. Final simplification 79.7%

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

Alternative 10: 79.3% accurate, 24.1× speedup

Math

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

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 67.7%

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

3. Taylor expanded in x around 0 55.0%

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

   1. associate-/r* 55.0%

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

   2. *-commutative 55.0%

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

   3. unpow2 55.0%

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

   4. unpow2 55.0%

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

   5. swap-sqr 68.2%

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

   6. unpow2 68.2%

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

   7. associate-/r* 68.9%

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

   8. unpow2 68.9%

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

   9. unpow2 68.9%

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

   10. swap-sqr 79.7%

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

   11. unpow2 79.7%

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

   12. *-commutative 79.7%

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

5. Simplified 79.7%

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

   1. *-commutative 79.7%

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

   2. unpow2 79.7%

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

   3. associate-*l* 77.6%

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

7. Applied egg-rr 77.6%

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

8. Final simplification 77.6%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))