mixedcos

Percentage Accurate: 67.3% → 99.4%

Time: 14.2s

Alternatives: 7

Speedup: 24.1×

• 31.8% 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: 67.3% 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: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{1}{{\left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}^{-2}}}\\ \end{array} \end{array} \]

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))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 5e-26)
     (/ (/ t_0 t_1) t_1)
     (/ t_0 (/ 1.0 (pow (* s_m (* x_m c_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) {
	double t_0 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5e-26) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 / (1.0 / pow((s_m * (x_m * c_m)), -2.0));
	}
	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 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 5d-26) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = t_0 / (1.0d0 / ((s_m * (x_m * c_m)) ** (-2.0d0)))
    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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5e-26) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 / (1.0 / Math.pow((s_m * (x_m * c_m)), -2.0));
	}
	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 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 5e-26:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = t_0 / (1.0 / math.pow((s_m * (x_m * c_m)), -2.0))
	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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 5e-26)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(t_0 / Float64(1.0 / (Float64(s_m * Float64(x_m * c_m)) ^ -2.0)));
	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 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 5e-26)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = t_0 / (1.0 / ((s_m * (x_m * c_m)) ^ -2.0));
	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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5e-26], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / N[(1.0 / N[Power[N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000019e-26

   1. Initial program 66.7%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. associate-/r* 61.5%

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

      2. *-commutative 61.5%

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

      3. unpow2 61.5%

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

      4. unpow2 61.5%

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

      5. swap-sqr 76.6%

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

      6. unpow2 76.6%

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

      7. associate-/r* 76.7%

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

      8. unpow2 76.7%

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

      9. unpow2 76.7%

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

      10. swap-sqr 97.1%

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

      11. unpow2 97.1%

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

      12. *-commutative 97.1%

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

   5. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. *-un-lft-identity 97.1%

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

      3. unpow2 97.1%

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

      4. times-frac 97.1%

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

      5. *-commutative 97.1%

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

      6. *-commutative 97.1%

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

      7. associate-*l* 95.3%

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

      8. *-commutative 95.3%

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

      9. *-commutative 95.3%

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

      10. *-commutative 95.3%

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

      11. associate-*l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. frac-times 97.6%

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

      2. *-un-lft-identity 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-/r* 97.7%

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

      5. associate-*r* 95.3%

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

      6. *-commutative 95.3%

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

      7. *-commutative 95.3%

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

      8. associate-*r* 97.1%

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

      9. *-commutative 97.1%

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

   9. Applied egg-rr 97.1%

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

   if 5.00000000000000019e-26 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 62.6%

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

      1. associate-/r* 62.6%

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

      2. *-commutative 62.6%

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

      3. unpow2 62.6%

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

      4. unpow2 62.6%

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

      5. swap-sqr 75.2%

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

      6. unpow2 75.2%

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

      7. associate-/r* 75.2%

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

      8. unpow2 75.2%

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

      9. unpow2 75.2%

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

      10. swap-sqr 96.7%

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

      11. unpow2 96.7%

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

      12. *-commutative 96.7%

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

   5. Simplified 96.7%

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

      1. /-rgt-identity 96.7%

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

      2. clear-num 96.6%

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

      3. pow-flip 96.6%

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

      4. *-commutative 96.6%

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

      5. *-commutative 96.6%

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

      6. associate-*l* 96.5%

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

      7. metadata-eval 96.5%

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

   7. Applied egg-rr 96.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ t_2 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m)))
        (t_1 (cos (* x_m 2.0)))
        (t_2 (* c_m (* x_m s_m))))
   (if (<= x_m 1e+17) (/ (/ t_1 t_2) t_2) (/ (/ t_1 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 = cos((x_m * 2.0));
	double t_2 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1e+17) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = (t_1 / 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) :: t_2
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * 2.0d0))
    t_2 = c_m * (x_m * s_m)
    if (x_m <= 1d+17) then
        tmp = (t_1 / t_2) / t_2
    else
        tmp = (t_1 / 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 = Math.cos((x_m * 2.0));
	double t_2 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1e+17) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = (t_1 / 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 = math.cos((x_m * 2.0))
	t_2 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1e+17:
		tmp = (t_1 / t_2) / t_2
	else:
		tmp = (t_1 / 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 = cos(Float64(x_m * 2.0))
	t_2 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1e+17)
		tmp = Float64(Float64(t_1 / t_2) / t_2);
	else
		tmp = Float64(Float64(t_1 / 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 = cos((x_m * 2.0));
	t_2 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1e+17)
		tmp = (t_1 / t_2) / t_2;
	else
		tmp = (t_1 / 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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1e+17], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e17

   1. Initial program 66.6%

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

   3. Taylor expanded in x around inf 61.7%

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

      1. associate-/r* 61.6%

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

      2. *-commutative 61.6%

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

      3. unpow2 61.6%

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

      4. unpow2 61.6%

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

      5. swap-sqr 76.1%

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

      6. unpow2 76.1%

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

      7. associate-/r* 76.2%

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

      8. unpow2 76.2%

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

      9. unpow2 76.2%

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

      10. swap-sqr 97.2%

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

      11. unpow2 97.2%

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

      12. *-commutative 97.2%

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

   5. Simplified 97.2%

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

      1. *-commutative 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. unpow2 97.2%

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

      4. times-frac 97.2%

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

      5. *-commutative 97.2%

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

      6. *-commutative 97.2%

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

      7. associate-*l* 95.5%

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

      8. *-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. *-commutative 95.5%

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

      11. associate-*l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. frac-times 97.7%

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

      2. *-un-lft-identity 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-/r* 97.8%

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

      5. associate-*r* 95.5%

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

      6. *-commutative 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-*r* 97.2%

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

      9. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 1e17 < x

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

      2. cos-neg 66.1%

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

      3. distribute-rgt-neg-out 66.1%

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

      4. distribute-rgt-neg-out 66.1%

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

      5. *-commutative 66.1%

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

      6. distribute-rgt-neg-in 66.1%

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

      7. metadata-eval 66.1%

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

      8. *-commutative 66.1%

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

      9. associate-*l* 62.4%

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

      10. unpow2 62.4%

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

   3. Simplified 62.4%

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

   6. Applied egg-rr 96.2%

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

      1. associate-*l/ 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. unpow2 96.2%

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

      4. associate-/r* 96.2%

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

      5. *-commutative 96.2%

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

      6. *-commutative 96.2%

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

      7. *-commutative 96.2%

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

      8. associate-*l* 92.8%

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

      9. *-commutative 92.8%

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

      10. *-commutative 92.8%

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

      11. associate-*l* 96.1%

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

   8. Applied egg-rr 96.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 1.4e+36)
     (/ (/ t_0 t_1) t_1)
     (/ 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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.4e+36) {
		tmp = (t_0 / t_1) / t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 1.4d+36) then
        tmp = (t_0 / t_1) / t_1
    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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.4e+36) {
		tmp = (t_0 / t_1) / t_1;
	} 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))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.4e+36:
		tmp = (t_0 / t_1) / t_1
	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))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.4e+36)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.4e+36)
		tmp = (t_0 / t_1) / t_1;
	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]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.4e+36], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e36

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 61.4%

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

      1. associate-/r* 61.3%

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

      2. *-commutative 61.3%

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

      3. unpow2 61.3%

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

      4. unpow2 61.3%

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

      5. swap-sqr 75.8%

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

      6. unpow2 75.8%

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

      7. associate-/r* 75.8%

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

      8. unpow2 75.8%

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

      9. unpow2 75.8%

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

      10. swap-sqr 97.2%

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

      11. unpow2 97.2%

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

      12. *-commutative 97.2%

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

   5. Simplified 97.2%

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

      1. *-commutative 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. unpow2 97.2%

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

      4. times-frac 97.2%

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

      5. *-commutative 97.2%

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

      6. *-commutative 97.2%

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

      7. associate-*l* 95.5%

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

      8. *-commutative 95.5%

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

      9. *-commutative 95.5%

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

      10. *-commutative 95.5%

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

      11. associate-*l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. frac-times 97.7%

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

      2. *-un-lft-identity 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-/r* 97.8%

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

      5. associate-*r* 95.5%

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

      6. *-commutative 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-*r* 97.2%

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

      9. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 1.4e36 < x

   1. Initial program 67.1%

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

   3. Taylor expanded in x around inf 63.3%

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

      1. associate-/r* 63.3%

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

      2. *-commutative 63.3%

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

      3. unpow2 63.3%

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

      4. unpow2 63.3%

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

      5. swap-sqr 78.0%

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

      6. unpow2 78.0%

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

      7. associate-/r* 78.0%

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

      8. unpow2 78.0%

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

      9. unpow2 78.0%

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

      10. swap-sqr 96.2%

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

      11. unpow2 96.2%

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

      12. *-commutative 96.2%

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

   5. Simplified 96.2%

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

      1. unpow2 96.2%

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

      2. associate-*r* 92.7%

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

      3. associate-*r* 89.3%

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

      4. *-commutative 89.3%

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

      5. *-commutative 89.3%

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

      6. associate-*l* 92.8%

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

      7. *-commutative 92.8%

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

   7. Applied egg-rr 92.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.1 \cdot 10^{-8}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 4.1e-8)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (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 tmp;
	if (x_m <= 4.1e-8) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} 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) :: tmp
    if (x_m <= 4.1d-8) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4.1e-8) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} 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):
	tmp = 0
	if x_m <= 4.1e-8:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	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)
	tmp = 0.0
	if (x_m <= 4.1e-8)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 4.1e-8)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	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_] := If[LessEqual[x$95$m, 4.1e-8], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.10000000000000032e-8

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. cos-neg 66.2%

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

      3. distribute-rgt-neg-out 66.2%

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

      4. distribute-rgt-neg-out 66.2%

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

      5. *-commutative 66.2%

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

      6. distribute-rgt-neg-in 66.2%

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

      7. metadata-eval 66.2%

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

      8. *-commutative 66.2%

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

      9. associate-*l* 61.1%

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

      10. unpow2 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in x around 0 59.6%

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

      1. associate-/r* 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

      4. unpow2 59.5%

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

      5. swap-sqr 70.5%

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

      6. unpow2 70.5%

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

      7. associate-/r* 70.6%

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

      8. unpow2 70.6%

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

      9. unpow2 70.6%

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

      10. swap-sqr 87.6%

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

      11. unpow2 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. pow-flip 87.6%

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

      3. *-commutative 87.6%

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

      4. *-commutative 87.6%

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

      5. associate-*l* 87.8%

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

      6. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. *-lft-identity 87.8%

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

       2. associate-*r* 87.6%

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

       3. *-commutative 87.6%

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

       4. *-commutative 87.6%

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

       5. *-commutative 87.6%

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

   11. Simplified 87.6%

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

   if 4.10000000000000032e-8 < x

   1. Initial program 67.4%

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

   3. Taylor expanded in x around inf 64.0%

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

      1. associate-/r* 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

      4. unpow2 64.0%

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

      5. swap-sqr 77.2%

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

      6. unpow2 77.2%

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

      7. associate-/r* 77.2%

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

      8. unpow2 77.2%

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

      9. unpow2 77.2%

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

      10. swap-sqr 96.5%

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

      11. unpow2 96.5%

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

      12. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. unpow2 96.5%

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

      2. associate-*r* 93.4%

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

      3. associate-*r* 90.4%

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

      4. *-commutative 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 93.5%

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

      7. *-commutative 93.5%

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

   7. Applied egg-rr 93.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{\left(c\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right) \cdot \left(x\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.4e-7)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m 2.0)) (* (* c_m (* s_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 <= 1.4e-7) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * 2.0)) / ((c_m * (s_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 <= 1.4d-7) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / ((c_m * (s_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 <= 1.4e-7) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / ((c_m * (s_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 <= 1.4e-7:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / ((c_m * (s_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 <= 1.4e-7)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(Float64(c_m * Float64(s_m * Float64(x_m * c_m))) * Float64(x_m * s_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.4e-7)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * 2.0)) / ((c_m * (s_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, 1.4e-7], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4000000000000001e-7

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. cos-neg 66.2%

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

      3. distribute-rgt-neg-out 66.2%

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

      4. distribute-rgt-neg-out 66.2%

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

      5. *-commutative 66.2%

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

      6. distribute-rgt-neg-in 66.2%

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

      7. metadata-eval 66.2%

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

      8. *-commutative 66.2%

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

      9. associate-*l* 61.1%

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

      10. unpow2 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in x around 0 59.6%

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

      1. associate-/r* 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

      4. unpow2 59.5%

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

      5. swap-sqr 70.5%

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

      6. unpow2 70.5%

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

      7. associate-/r* 70.6%

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

      8. unpow2 70.6%

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

      9. unpow2 70.6%

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

      10. swap-sqr 87.6%

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

      11. unpow2 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. pow-flip 87.6%

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

      3. *-commutative 87.6%

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

      4. *-commutative 87.6%

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

      5. associate-*l* 87.8%

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

      6. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. *-lft-identity 87.8%

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

       2. associate-*r* 87.6%

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

       3. *-commutative 87.6%

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

       4. *-commutative 87.6%

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

       5. *-commutative 87.6%

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

   11. Simplified 87.6%

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

   if 1.4000000000000001e-7 < x

   1. Initial program 67.4%

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

   3. Taylor expanded in x around inf 64.0%

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

      1. associate-/r* 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

      4. unpow2 64.0%

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

      5. swap-sqr 77.2%

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

      6. unpow2 77.2%

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

      7. associate-/r* 77.2%

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

      8. unpow2 77.2%

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

      9. unpow2 77.2%

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

      10. swap-sqr 96.5%

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

      11. unpow2 96.5%

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

      12. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. unpow2 96.5%

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

      2. associate-*r* 91.5%

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

      3. *-commutative 91.5%

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

      4. *-commutative 91.5%

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

      5. associate-*l* 89.9%

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% 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 66.5%

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

   1. associate-/r* 66.5%

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

   2. cos-neg 66.5%

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

   3. distribute-rgt-neg-out 66.5%

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

   4. distribute-rgt-neg-out 66.5%

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

   5. *-commutative 66.5%

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

   6. distribute-rgt-neg-in 66.5%

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

   7. metadata-eval 66.5%

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

   8. *-commutative 66.5%

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

   9. associate-*l* 61.8%

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

   10. unpow2 61.8%

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

3. Simplified 61.8%

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

5. Taylor expanded in x around 0 57.8%

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

   1. associate-/r* 57.8%

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

   2. *-commutative 57.8%

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

   3. unpow2 57.8%

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

   4. unpow2 57.8%

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

   5. swap-sqr 67.6%

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

   6. unpow2 67.6%

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

   7. associate-/r* 67.6%

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

   8. unpow2 67.6%

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

   9. unpow2 67.6%

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

   10. swap-sqr 81.9%

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

   11. unpow2 81.9%

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

7. Simplified 81.9%

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

   1. *-un-lft-identity 81.9%

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

   2. pow-flip 81.9%

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

   3. *-commutative 81.9%

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

   4. *-commutative 81.9%

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

   5. associate-*l* 82.0%

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

   6. metadata-eval 82.0%

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

9. Applied egg-rr 82.0%

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

    1. *-lft-identity 82.0%

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

    2. associate-*r* 81.9%

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

    3. *-commutative 81.9%

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

    4. *-commutative 81.9%

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

    5. *-commutative 81.9%

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

11. Simplified 81.9%

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

12. Final simplification 81.9%

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

Alternative 7: 80.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. associate-/r* 66.5%

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

   2. cos-neg 66.5%

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

   3. distribute-rgt-neg-out 66.5%

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

   4. distribute-rgt-neg-out 66.5%

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

   5. *-commutative 66.5%

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

   6. distribute-rgt-neg-in 66.5%

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

   7. metadata-eval 66.5%

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

   8. *-commutative 66.5%

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

   9. associate-*l* 61.8%

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

   10. unpow2 61.8%

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

3. Simplified 61.8%

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

5. Taylor expanded in x around 0 57.8%

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

   1. associate-/r* 57.8%

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

   2. *-commutative 57.8%

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

   3. unpow2 57.8%

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

   4. unpow2 57.8%

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

   5. swap-sqr 67.6%

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

   6. unpow2 67.6%

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

   7. associate-/r* 67.6%

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

   8. unpow2 67.6%

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

   9. unpow2 67.6%

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

   10. swap-sqr 81.9%

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

   11. unpow2 81.9%

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

7. Simplified 81.9%

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

   1. add-sqr-sqrt 81.9%

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

   2. sqrt-unprod 74.4%

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

   3. pow-prod-up 74.4%

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

   4. *-commutative 74.4%

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

   5. *-commutative 74.4%

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

   6. associate-*l* 74.4%

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

   7. metadata-eval 74.4%

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

9. Applied egg-rr 74.4%

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

    1. associate-*r* 74.4%

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

    2. *-commutative 74.4%

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

    3. *-commutative 74.4%

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

    4. *-commutative 74.4%

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

11. Simplified 74.4%

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

    1. sqrt-pow1 81.9%

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

    2. metadata-eval 81.9%

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

    3. unpow-prod-down 67.6%

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

    4. pow2 67.6%

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

    5. pow2 67.6%

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

    6. swap-sqr 81.9%

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

13. Applied egg-rr 81.9%

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

14. Final simplification 81.9%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))