mixedcos

Percentage Accurate: 65.8% → 98.2%

Time: 20.5s

Alternatives: 10

Speedup: 24.1×

• 31.3% 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: 65.8% 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: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{s\_m \cdot \left(\left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right) \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 1.7e-50)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m -2.0)) (* s_m (* (* s_m (* x_m c_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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.7e-50) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * ((s_m * (x_m * c_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 = c_m * (x_m * s_m)
    if (x_m <= 1.7d-50) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * ((s_m * (x_m * c_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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.7e-50) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * ((s_m * (x_m * c_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 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.7e-50:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * ((s_m * (x_m * c_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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.7e-50)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(s_m * Float64(x_m * c_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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.7e-50)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * ((s_m * (x_m * c_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.7e-50], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.70000000000000007e-50

   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. Add Preprocessing

   3. Taylor expanded in x around 0 55.8%

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

      1. associate-/r* 55.8%

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

      2. *-commutative 55.8%

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

      3. unpow2 55.8%

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

      4. unpow2 55.8%

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

      5. swap-sqr 66.9%

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

      6. unpow2 66.9%

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

      7. associate-/r* 66.6%

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

      8. unpow2 66.6%

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

      9. unpow2 66.6%

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

      10. swap-sqr 83.6%

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

      11. unpow2 83.6%

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

      12. *-commutative 83.6%

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

   5. Simplified 83.6%

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

      1. *-commutative 83.6%

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

      2. add-sqr-sqrt 83.5%

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

      3. sqrt-div 83.5%

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

      4. metadata-eval 83.5%

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

      5. sqrt-pow1 58.7%

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

      6. associate-*r* 57.7%

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

      7. metadata-eval 57.7%

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

      8. pow1 57.7%

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

      9. *-commutative 57.7%

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

      10. sqrt-div 57.7%

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

      11. metadata-eval 57.7%

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

      12. sqrt-pow1 81.7%

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

      13. associate-*r* 81.8%

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

      14. metadata-eval 81.8%

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

      15. pow1 81.8%

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

      16. *-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. un-div-inv 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*l* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 83.9%

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

   9. Applied egg-rr 83.9%

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

   if 1.70000000000000007e-50 < x

   1. Initial program 66.9%

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

      1. associate-/r* 65.2%

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

      2. associate-/l/ 66.8%

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

      3. unpow2 66.8%

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

      4. sqr-neg 66.8%

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

      5. unpow2 66.8%

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

      6. associate-/r* 65.2%

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

      7. *-commutative 65.2%

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

      8. associate-/r* 66.9%

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

      9. cos-neg 66.9%

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

      10. *-commutative 66.9%

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

      11. distribute-rgt-neg-in 66.9%

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

      12. metadata-eval 66.9%

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

      13. *-commutative 66.9%

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

      14. associate-*r* 66.8%

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

      15. unpow2 66.8%

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

      16. sqr-neg 66.8%

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

      17. unpow2 66.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around inf 58.8%

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

      1. associate-/r* 57.0%

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

      2. *-commutative 57.0%

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

      3. unpow2 57.0%

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

      4. unpow2 57.0%

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

      5. swap-sqr 70.8%

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

      6. unpow2 70.8%

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

      7. associate-/r* 72.6%

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

      8. *-commutative 72.6%

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

      9. unpow2 72.6%

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

      10. unpow2 72.6%

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

      11. swap-sqr 98.3%

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

      12. unpow2 98.3%

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

      13. associate-*r* 95.9%

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

      14. *-commutative 95.9%

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

   7. Simplified 95.9%

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

      1. unpow2 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-*r* 94.7%

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

      4. *-commutative 94.7%

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

      5. associate-*r* 92.1%

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

      6. associate-*r* 93.4%

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

      7. *-commutative 93.4%

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

   9. Applied egg-rr 93.4%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{s\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.2e-56)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* s_m (* c_m (* x_m (* x_m (* s_m c_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.2e-56) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * (c_m * (x_m * (x_m * (s_m * c_m)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.2d-56) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * (c_m * (x_m * (x_m * (s_m * c_m)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.2e-56) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * (c_m * (x_m * (x_m * (s_m * c_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2.2e-56:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * (c_m * (x_m * (x_m * (s_m * c_m)))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2.2e-56)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(c_m * Float64(x_m * Float64(x_m * Float64(s_m * c_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2.2e-56)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * (c_m * (x_m * (x_m * (s_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2.2e-56], 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[(c$95$m * N[(x$95$m * N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000004e-56

   1. Initial program 66.9%

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

      1. associate-/r* 66.8%

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

      2. associate-/l/ 68.1%

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

      3. unpow2 68.1%

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

      4. sqr-neg 68.1%

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

      5. unpow2 68.1%

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

      6. associate-/r* 66.8%

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

      7. *-commutative 66.8%

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

      8. associate-/r* 66.9%

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

      9. cos-neg 66.9%

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

      10. *-commutative 66.9%

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

      11. distribute-rgt-neg-in 66.9%

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

      12. metadata-eval 66.9%

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

      13. *-commutative 66.9%

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

      14. associate-*r* 67.7%

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

      15. unpow2 67.7%

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

      16. sqr-neg 67.7%

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

      17. unpow2 67.7%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in x around inf 60.4%

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

      1. associate-/r* 60.4%

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

      2. *-commutative 60.4%

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

      3. unpow2 60.4%

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

      4. unpow2 60.4%

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

      5. swap-sqr 74.8%

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

      6. unpow2 74.8%

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

      7. associate-/r* 74.5%

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

      8. *-commutative 74.5%

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

      9. unpow2 74.5%

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

      10. unpow2 74.5%

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

      11. swap-sqr 98.1%

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

      12. unpow2 98.1%

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

      13. associate-*r* 95.5%

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

      14. *-commutative 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in x around 0 56.2%

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

      1. associate-*r* 56.0%

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

      2. *-commutative 56.0%

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

      3. associate-*l* 56.4%

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

      4. unpow2 56.4%

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

      5. unpow2 56.4%

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

      6. swap-sqr 72.8%

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

      7. unpow2 72.8%

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

      8. swap-sqr 81.3%

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

      9. associate-/r* 81.7%

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

      10. *-rgt-identity 81.7%

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

      11. associate-*r/ 81.7%

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

      12. unpow-1 81.7%

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

      13. unpow-1 81.7%

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

      14. pow-sqr 81.7%

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

      15. metadata-eval 81.7%

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

      16. associate-*r* 82.4%

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

      17. *-commutative 82.4%

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

      18. associate-*r* 83.9%

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

   10. Simplified 83.9%

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

   if 2.20000000000000004e-56 < x

   1. Initial program 66.0%

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

      1. associate-/r* 64.3%

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

      2. associate-/l/ 65.9%

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

      3. unpow2 65.9%

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

      4. sqr-neg 65.9%

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

      5. unpow2 65.9%

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

      6. associate-/r* 64.3%

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

      7. *-commutative 64.3%

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

      8. associate-/r* 66.0%

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

      9. cos-neg 66.0%

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

      10. *-commutative 66.0%

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

      11. distribute-rgt-neg-in 66.0%

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

      12. metadata-eval 66.0%

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

      13. *-commutative 66.0%

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

      14. associate-*r* 65.9%

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

      15. unpow2 65.9%

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

      16. sqr-neg 65.9%

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

      17. unpow2 65.9%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in x around inf 58.0%

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

      1. associate-/r* 56.3%

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

      2. *-commutative 56.3%

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

      3. unpow2 56.3%

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

      4. unpow2 56.3%

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

      5. swap-sqr 69.8%

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

      6. unpow2 69.8%

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

      7. associate-/r* 71.6%

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

      8. *-commutative 71.6%

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

      9. unpow2 71.6%

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

      10. unpow2 71.6%

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

      11. swap-sqr 98.3%

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

      12. unpow2 98.3%

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

      13. associate-*r* 95.9%

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

      14. *-commutative 95.9%

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

   7. Simplified 95.9%

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

      1. add0 95.9%

         \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(s \cdot \left(c \cdot x\right) + 0\right)}}^{2}} \]

      2. associate-*r* 97.2%

         \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\left(\color{blue}{\left(s \cdot c\right) \cdot x} + 0\right)}^{2}} \]

      3. fma-define 97.2%

         \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\mathsf{fma}\left(s \cdot c, x, 0\right)\right)}}^{2}} \]

   9. Applied egg-rr 97.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\mathsf{fma}\left(s \cdot c, x, 0\right)\right)}}^{2}} \]
   10. Step-by-step derivation

       1. fma-undefine 97.2%

          \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\left(s \cdot c\right) \cdot x + 0\right)}}^{2}} \]

       2. add0 97.2%

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

   11. Applied egg-rr 97.2%

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

       1. unpow2 97.2%

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

       2. associate-*r* 93.6%

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

       3. *-commutative 93.6%

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

       4. associate-*r* 95.9%

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

       5. associate-*r* 91.0%

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

       6. associate-*l* 91.1%

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

       7. associate-*r* 92.3%

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

       8. *-commutative 92.3%

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

       9. associate-*r* 89.9%

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

       10. *-commutative 89.9%

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

   13. Applied egg-rr 89.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 (* c_m (* x_m s_m))))
   (if (<= x_m 4e-56)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m -2.0)) (* (* s_m c_m) (* x_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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4e-56) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_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 = c_m * (x_m * s_m)
    if (x_m <= 4d-56) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * (-2.0d0))) / ((s_m * c_m) * (x_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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4e-56) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_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 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 4e-56:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 4e-56)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(s_m * c_m) * Float64(x_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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 4e-56)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4e-56], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * 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 < 4.0000000000000002e-56

   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. Add Preprocessing

   3. Taylor expanded in x around 0 55.8%

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

      1. associate-/r* 55.8%

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

      2. *-commutative 55.8%

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

      3. unpow2 55.8%

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

      4. unpow2 55.8%

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

      5. swap-sqr 66.9%

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

      6. unpow2 66.9%

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

      7. associate-/r* 66.6%

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

      8. unpow2 66.6%

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

      9. unpow2 66.6%

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

      10. swap-sqr 83.6%

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

      11. unpow2 83.6%

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

      12. *-commutative 83.6%

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

   5. Simplified 83.6%

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

      1. *-commutative 83.6%

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

      2. add-sqr-sqrt 83.5%

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

      3. sqrt-div 83.5%

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

      4. metadata-eval 83.5%

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

      5. sqrt-pow1 58.7%

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

      6. associate-*r* 57.7%

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

      7. metadata-eval 57.7%

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

      8. pow1 57.7%

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

      9. *-commutative 57.7%

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

      10. sqrt-div 57.7%

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

      11. metadata-eval 57.7%

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

      12. sqrt-pow1 81.7%

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

      13. associate-*r* 81.8%

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

      14. metadata-eval 81.8%

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

      15. pow1 81.8%

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

      16. *-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. un-div-inv 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*l* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 83.9%

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

   9. Applied egg-rr 83.9%

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

   if 4.0000000000000002e-56 < x

   1. Initial program 66.9%

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

      1. associate-/r* 65.2%

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

      2. associate-/l/ 66.8%

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

      3. unpow2 66.8%

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

      4. sqr-neg 66.8%

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

      5. unpow2 66.8%

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

      6. associate-/r* 65.2%

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

      7. *-commutative 65.2%

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

      8. associate-/r* 66.9%

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

      9. cos-neg 66.9%

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

      10. *-commutative 66.9%

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

      11. distribute-rgt-neg-in 66.9%

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

      12. metadata-eval 66.9%

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

      13. *-commutative 66.9%

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

      14. associate-*r* 66.8%

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

      15. unpow2 66.8%

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

      16. sqr-neg 66.8%

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

      17. unpow2 66.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around inf 58.8%

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

      1. associate-/r* 57.0%

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

      2. *-commutative 57.0%

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

      3. unpow2 57.0%

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

      4. unpow2 57.0%

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

      5. swap-sqr 70.8%

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

      6. unpow2 70.8%

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

      7. associate-/r* 72.6%

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

      8. *-commutative 72.6%

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

      9. unpow2 72.6%

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

      10. unpow2 72.6%

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

      11. swap-sqr 98.3%

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

      12. unpow2 98.3%

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

      13. associate-*r* 95.9%

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

      14. *-commutative 95.9%

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

   7. Simplified 95.9%

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

      1. unpow2 95.9%

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

      2. associate-*r* 93.5%

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

      3. *-commutative 93.5%

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

      4. associate-*r* 95.9%

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

      5. associate-*l* 90.9%

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

      6. associate-*r* 90.9%

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

      7. *-commutative 90.9%

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s\_m \cdot \left(s\_m \cdot {\left(x\_m \cdot c\_m\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 2.9e+22)
     (/ (/ 1.0 t_0) t_0)
     (/ -1.0 (* s_m (* s_m (pow (* 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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.9e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = -1.0 / (s_m * (s_m * pow((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) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 2.9d+22) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (-1.0d0) / (s_m * (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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.9e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = -1.0 / (s_m * (s_m * Math.pow((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 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 2.9e+22:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = -1.0 / (s_m * (s_m * math.pow((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 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 2.9e+22)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(-1.0 / Float64(s_m * 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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.9e+22)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = -1.0 / (s_m * (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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.9e+22], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 / N[(s$95$m * N[(s$95$m * N[Power[N[(x$95$m * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e22

   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 0 56.2%

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

      1. associate-/r* 55.5%

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

      2. *-commutative 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 66.3%

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

      8. unpow2 66.3%

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

      9. unpow2 66.3%

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

      10. swap-sqr 83.4%

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

      11. unpow2 83.4%

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

      12. *-commutative 83.4%

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

   5. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. add-sqr-sqrt 83.4%

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

      3. sqrt-div 83.4%

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

      4. metadata-eval 83.4%

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

      5. sqrt-pow1 59.7%

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

      6. associate-*r* 58.8%

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

      7. metadata-eval 58.8%

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

      8. pow1 58.8%

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

      9. *-commutative 58.8%

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

      10. sqrt-div 58.8%

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

      11. metadata-eval 58.8%

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

      12. sqrt-pow1 81.7%

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

      13. associate-*r* 81.8%

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

      14. metadata-eval 81.8%

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

      15. pow1 81.8%

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

      16. *-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. un-div-inv 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*l* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 83.8%

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

   9. Applied egg-rr 83.8%

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

   if 2.9e22 < x

   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 0 45.8%

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

      1. associate-/r* 45.8%

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

      2. *-commutative 45.8%

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

      3. unpow2 45.8%

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

      4. unpow2 45.8%

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

      5. swap-sqr 50.3%

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

      6. unpow2 50.3%

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

      7. associate-/r* 50.3%

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

      8. unpow2 50.3%

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

      9. unpow2 50.3%

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

      10. swap-sqr 59.1%

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

      11. unpow2 59.1%

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

      12. *-commutative 59.1%

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

   5. Simplified 59.1%

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

      1. *-commutative 59.1%

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

      2. add-sqr-sqrt 59.1%

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

      3. sqrt-div 59.1%

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

      4. metadata-eval 59.1%

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

      5. sqrt-pow1 59.7%

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

      6. associate-*r* 59.3%

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

      7. metadata-eval 59.3%

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

      8. pow1 59.3%

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

      9. *-commutative 59.3%

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

      10. sqrt-div 59.3%

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

      11. metadata-eval 59.3%

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

      12. sqrt-pow1 59.0%

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

      13. associate-*r* 59.0%

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

      14. metadata-eval 59.0%

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

      15. pow1 59.0%

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

      16. *-commutative 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. associate-/r* 59.0%

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

      2. frac-times 58.8%

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

      3. *-un-lft-identity 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*l* 58.8%

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

      6. *-commutative 58.8%

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

   9. Applied egg-rr 58.8%

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

   11. Applied egg-rr 63.4%

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

       1. associate-*r/ 63.4%

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

       2. metadata-eval 63.4%

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

       3. *-commutative 63.4%

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

       4. *-commutative 63.4%

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

   13. Simplified 63.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(s \cdot {\left(x \cdot c\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 2.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* 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(x_m * Float64(s_m * c_m))
	return Float64(cos(Float64(x_m * -2.0)) / 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 = 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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

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. 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. associate-/l/ 67.4%

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

   3. unpow2 67.4%

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

   4. sqr-neg 67.4%

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

   5. unpow2 67.4%

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

   6. associate-/r* 66.1%

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

   7. *-commutative 66.1%

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

   8. associate-/r* 66.6%

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

   9. cos-neg 66.6%

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

   10. *-commutative 66.6%

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

   11. distribute-rgt-neg-in 66.6%

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

   12. metadata-eval 66.6%

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

   13. *-commutative 66.6%

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

   14. associate-*r* 67.2%

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

   15. unpow2 67.2%

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

   16. sqr-neg 67.2%

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

   17. unpow2 67.2%

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

3. Simplified 59.7%

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

5. Taylor expanded in x around inf 59.7%

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

   1. associate-/r* 59.2%

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

   2. *-commutative 59.2%

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

   3. unpow2 59.2%

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

   4. unpow2 59.2%

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

   5. swap-sqr 73.4%

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

   6. unpow2 73.4%

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

   7. associate-/r* 73.6%

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

   8. *-commutative 73.6%

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

   9. unpow2 73.6%

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

   10. unpow2 73.6%

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

   11. swap-sqr 98.2%

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

   12. unpow2 98.2%

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

   13. associate-*r* 95.7%

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

   14. *-commutative 95.7%

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

7. Simplified 95.7%

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

   1. add0 95.7%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(s \cdot \left(c \cdot x\right) + 0\right)}}^{2}} \]

   2. associate-*r* 97.1%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\left(\color{blue}{\left(s \cdot c\right) \cdot x} + 0\right)}^{2}} \]

   3. fma-define 97.1%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\mathsf{fma}\left(s \cdot c, x, 0\right)\right)}}^{2}} \]

9. Applied egg-rr 97.1%

   \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\mathsf{fma}\left(s \cdot c, x, 0\right)\right)}}^{2}} \]
10. Step-by-step derivation

    1. fma-undefine 97.1%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\color{blue}{\left(\left(s \cdot c\right) \cdot x + 0\right)}}^{2}} \]

    2. add0 97.1%

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

11. Applied egg-rr 97.1%

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

    1. unpow2 97.1%

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

    2. *-commutative 97.1%

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

    3. *-commutative 97.1%

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

13. Applied egg-rr 97.1%

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

14. Final simplification 97.1%

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

Alternative 6: 97.3% accurate, 2.7× speedup

Math

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

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)))) (/ (/ (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);
	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 = s_m * (x_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 = s_m * (x_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 = s_m * (x_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(s_m * Float64(x_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 = s_m * (x_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[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. 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. Step-by-step derivation

   1. div-inv 66.6%

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

   2. *-commutative 66.6%

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

   3. *-commutative 66.6%

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

   4. associate-*r* 59.7%

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

   5. unpow2 59.7%

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

   6. *-commutative 59.7%

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

   7. pow-prod-down 73.6%

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

   8. pow-prod-down 98.1%

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

   9. *-commutative 98.1%

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

4. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. unpow2 98.2%

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

   4. associate-/r* 98.4%

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

   5. add-sqr-sqrt 48.2%

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

   6. sqrt-unprod 71.9%

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

   7. *-commutative 71.9%

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

   8. *-commutative 71.9%

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

   9. swap-sqr 71.9%

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

   10. metadata-eval 71.9%

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

   11. metadata-eval 71.9%

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

   12. swap-sqr 71.9%

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

   13. sqrt-unprod 41.6%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-*r* 94.9%

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

   16. *-commutative 94.9%

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

   17. associate-*r* 95.9%

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

   18. *-commutative 95.9%

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

6. Applied egg-rr 95.9%

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

7. Final simplification 95.9%

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

Alternative 7: 78.9% accurate, 16.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s\_m}}{\left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(c\_m \cdot \left(-x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 2.9e+22)
     (/ (/ 1.0 t_0) t_0)
     (/ (/ 1.0 s_m) (* (* x_m (* s_m c_m)) (* c_m (- x_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.9e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / s_m) / ((x_m * (s_m * c_m)) * (c_m * -x_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 = c_m * (x_m * s_m)
    if (x_m <= 2.9d+22) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (1.0d0 / s_m) / ((x_m * (s_m * c_m)) * (c_m * -x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.9e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / s_m) / ((x_m * (s_m * c_m)) * (c_m * -x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 2.9e+22:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (1.0 / s_m) / ((x_m * (s_m * c_m)) * (c_m * -x_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 2.9e+22)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(1.0 / s_m) / Float64(Float64(x_m * Float64(s_m * c_m)) * Float64(c_m * Float64(-x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.9e+22)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (1.0 / s_m) / ((x_m * (s_m * c_m)) * (c_m * -x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.9e+22], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / s$95$m), $MachinePrecision] / N[(N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * (-x$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e22

   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 0 56.2%

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

      1. associate-/r* 55.5%

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

      2. *-commutative 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 66.3%

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

      8. unpow2 66.3%

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

      9. unpow2 66.3%

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

      10. swap-sqr 83.4%

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

      11. unpow2 83.4%

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

      12. *-commutative 83.4%

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

   5. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. add-sqr-sqrt 83.4%

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

      3. sqrt-div 83.4%

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

      4. metadata-eval 83.4%

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

      5. sqrt-pow1 59.7%

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

      6. associate-*r* 58.8%

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

      7. metadata-eval 58.8%

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

      8. pow1 58.8%

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

      9. *-commutative 58.8%

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

      10. sqrt-div 58.8%

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

      11. metadata-eval 58.8%

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

      12. sqrt-pow1 81.7%

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

      13. associate-*r* 81.8%

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

      14. metadata-eval 81.8%

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

      15. pow1 81.8%

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

      16. *-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. un-div-inv 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*l* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 83.8%

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

   9. Applied egg-rr 83.8%

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

   if 2.9e22 < x

   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 0 45.8%

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

      1. associate-/r* 45.8%

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

      2. *-commutative 45.8%

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

      3. unpow2 45.8%

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

      4. unpow2 45.8%

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

      5. swap-sqr 50.3%

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

      6. unpow2 50.3%

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

      7. associate-/r* 50.3%

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

      8. unpow2 50.3%

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

      9. unpow2 50.3%

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

      10. swap-sqr 59.1%

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

      11. unpow2 59.1%

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

      12. *-commutative 59.1%

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

   5. Simplified 59.1%

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

      1. *-commutative 59.1%

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

      2. add-sqr-sqrt 59.1%

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

      3. sqrt-div 59.1%

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

      4. metadata-eval 59.1%

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

      5. sqrt-pow1 59.7%

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

      6. associate-*r* 59.3%

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

      7. metadata-eval 59.3%

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

      8. pow1 59.3%

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

      9. *-commutative 59.3%

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

      10. sqrt-div 59.3%

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

      11. metadata-eval 59.3%

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

      12. sqrt-pow1 59.0%

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

      13. associate-*r* 59.0%

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

      14. metadata-eval 59.0%

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

      15. pow1 59.0%

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

      16. *-commutative 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. associate-/r* 59.0%

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

      2. frac-times 58.8%

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

      3. *-un-lft-identity 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*l* 58.8%

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

      6. *-commutative 58.8%

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

   9. Applied egg-rr 58.8%

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

       1. associate-*r* 58.8%

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

       2. *-commutative 58.8%

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

       3. add-sqr-sqrt 24.2%

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

       4. sqrt-unprod 64.7%

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

       5. sqr-neg 64.7%

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

       6. sqrt-unprod 40.8%

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

       7. add-sqr-sqrt 63.7%

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

       8. distribute-rgt-neg-in 63.7%

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

       9. neg-sub0 63.7%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(0 - s \cdot \left(c \cdot x\right)\right)} \cdot \left(x \cdot c\right)} \]

       10. associate-*r* 58.6%

           \[\leadsto \frac{\frac{1}{s}}{\left(0 - \color{blue}{\left(s \cdot c\right) \cdot x}\right) \cdot \left(x \cdot c\right)} \]

       11. *-commutative 58.6%

           \[\leadsto \frac{\frac{1}{s}}{\left(0 - \color{blue}{x \cdot \left(s \cdot c\right)}\right) \cdot \left(x \cdot c\right)} \]

   11. Applied egg-rr 58.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(0 - x \cdot \left(s \cdot c\right)\right)} \cdot \left(x \cdot c\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 63.7%

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

       2. distribute-rgt-neg-in 63.7%

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

       3. distribute-lft-neg-in 63.7%

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

   13. Simplified 63.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(c \cdot \left(-x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 24.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

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 0 53.8%

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

   1. associate-/r* 53.3%

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

   2. *-commutative 53.3%

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

   3. unpow2 53.3%

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

   4. unpow2 53.3%

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

   5. swap-sqr 62.3%

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

   6. unpow2 62.3%

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

   7. associate-/r* 62.6%

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

   8. unpow2 62.6%

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

   9. unpow2 62.6%

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

   10. swap-sqr 77.8%

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

   11. unpow2 77.8%

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

   12. *-commutative 77.8%

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

5. Simplified 77.8%

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

   1. *-commutative 77.8%

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

   2. add-sqr-sqrt 77.8%

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

   3. sqrt-div 77.8%

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

   4. metadata-eval 77.8%

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

   5. sqrt-pow1 59.7%

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

   6. associate-*r* 58.9%

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

   7. metadata-eval 58.9%

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

   8. pow1 58.9%

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

   9. *-commutative 58.9%

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

   10. sqrt-div 58.9%

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

   11. metadata-eval 58.9%

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

   12. sqrt-pow1 76.5%

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

   13. associate-*r* 76.5%

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

   14. metadata-eval 76.5%

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

   15. pow1 76.5%

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

   16. *-commutative 76.5%

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

7. Applied egg-rr 76.5%

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

   1. associate-/r* 76.6%

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

   2. frac-times 75.5%

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

   3. *-un-lft-identity 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*l* 75.4%

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

   6. *-commutative 75.4%

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

9. Applied egg-rr 75.4%

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

10. Final simplification 75.4%

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

Alternative 9: 73.8% accurate, 24.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

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 0 53.8%

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

   1. associate-/r* 53.3%

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

   2. *-commutative 53.3%

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

   3. unpow2 53.3%

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

   4. unpow2 53.3%

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

   5. swap-sqr 62.3%

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

   6. unpow2 62.3%

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

   7. associate-/r* 62.6%

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

   8. unpow2 62.6%

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

   9. unpow2 62.6%

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

   10. swap-sqr 77.8%

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

   11. unpow2 77.8%

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

   12. *-commutative 77.8%

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

5. Simplified 77.8%

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

   1. *-commutative 77.8%

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

   2. add-sqr-sqrt 77.8%

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

   3. sqrt-div 77.8%

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

   4. metadata-eval 77.8%

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

   5. sqrt-pow1 59.7%

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

   6. associate-*r* 58.9%

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

   7. metadata-eval 58.9%

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

   8. pow1 58.9%

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

   9. *-commutative 58.9%

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

   10. sqrt-div 58.9%

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

   11. metadata-eval 58.9%

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

   12. sqrt-pow1 76.5%

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

   13. associate-*r* 76.5%

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

   14. metadata-eval 76.5%

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

   15. pow1 76.5%

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

   16. *-commutative 76.5%

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

7. Applied egg-rr 76.5%

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

   1. associate-/r* 76.6%

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

   2. frac-times 75.5%

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

   3. *-un-lft-identity 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*l* 75.4%

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

   6. *-commutative 75.4%

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

9. Applied egg-rr 75.4%

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

    1. associate-*r* 75.5%

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

    2. *-commutative 75.5%

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

    3. add0 66.5%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(s \cdot \left(c \cdot x\right) + 0\right)} \cdot \left(x \cdot c\right)} \]

    4. fma-define 70.8%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{fma}\left(s, c \cdot x, 0\right)} \cdot \left(x \cdot c\right)} \]

11. Applied egg-rr 70.8%

    \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{fma}\left(s, c \cdot x, 0\right)} \cdot \left(x \cdot c\right)} \]
12. Step-by-step derivation

    1. fma-undefine 66.5%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(s \cdot \left(c \cdot x\right) + 0\right)} \cdot \left(x \cdot c\right)} \]

    2. add0 75.5%

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

    3. *-commutative 75.5%

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

13. Simplified 75.5%

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

14. Final simplification 75.5%

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

Alternative 10: 78.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 0 53.8%

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

   1. associate-/r* 53.3%

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

   2. *-commutative 53.3%

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

   3. unpow2 53.3%

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

   4. unpow2 53.3%

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

   5. swap-sqr 62.3%

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

   6. unpow2 62.3%

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

   7. associate-/r* 62.6%

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

   8. unpow2 62.6%

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

   9. unpow2 62.6%

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

   10. swap-sqr 77.8%

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

   11. unpow2 77.8%

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

   12. *-commutative 77.8%

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

5. Simplified 77.8%

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

   1. *-commutative 77.8%

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

   2. add-sqr-sqrt 77.8%

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

   3. sqrt-div 77.8%

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

   4. metadata-eval 77.8%

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

   5. sqrt-pow1 59.7%

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

   6. associate-*r* 58.9%

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

   7. metadata-eval 58.9%

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

   8. pow1 58.9%

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

   9. *-commutative 58.9%

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

   10. sqrt-div 58.9%

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

   11. metadata-eval 58.9%

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

   12. sqrt-pow1 76.5%

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

   13. associate-*r* 76.5%

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

   14. metadata-eval 76.5%

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

   15. pow1 76.5%

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

   16. *-commutative 76.5%

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

7. Applied egg-rr 76.5%

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

   1. un-div-inv 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.5%

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

   4. *-commutative 76.5%

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

   5. associate-*l* 78.1%

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

9. Applied egg-rr 78.1%

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

10. Final simplification 78.1%

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))