mixedcos

Percentage Accurate: 68.0% → 99.6%

Time: 12.8s

Alternatives: 7

Speedup: 24.1×

• 32.4% 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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{1}{s\_m} \cdot \frac{t\_1}{x\_m}}{c\_m}}{c\_m \cdot \left|x\_m \cdot s\_m\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 1.2e-27)
     (/ (/ (* (/ 1.0 s_m) (/ t_1 x_m)) c_m) (* c_m (fabs (* x_m s_m))))
     (/ (/ t_1 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.2e-27) {
		tmp = (((1.0 / s_m) * (t_1 / x_m)) / c_m) / (c_m * fabs((x_m * s_m)));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 1.2d-27) then
        tmp = (((1.0d0 / s_m) * (t_1 / x_m)) / c_m) / (c_m * abs((x_m * s_m)))
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.2e-27) {
		tmp = (((1.0 / s_m) * (t_1 / x_m)) / c_m) / (c_m * Math.abs((x_m * s_m)));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 1.2e-27:
		tmp = (((1.0 / s_m) * (t_1 / x_m)) / c_m) / (c_m * math.fabs((x_m * s_m)))
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 1.2e-27)
		tmp = Float64(Float64(Float64(Float64(1.0 / s_m) * Float64(t_1 / x_m)) / c_m) / Float64(c_m * abs(Float64(x_m * s_m))));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 1.2e-27)
		tmp = (((1.0 / s_m) * (t_1 / x_m)) / c_m) / (c_m * abs((x_m * s_m)));
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 1.2e-27], N[(N[(N[(N[(1.0 / s$95$m), $MachinePrecision] * N[(t$95$1 / x$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * N[Abs[N[(x$95$m * s$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000001e-27

   1. Initial program 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. add-sqr-sqrt 67.1%

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

      3. times-frac 67.1%

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

      4. sqrt-prod 67.1%

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

      5. sqrt-pow1 45.6%

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

      6. metadata-eval 45.6%

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

      7. pow1 45.6%

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

      8. *-commutative 45.6%

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

      9. associate-*r* 42.8%

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

      10. unpow2 42.8%

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

      11. pow-prod-down 45.6%

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

      12. sqrt-prod 45.6%

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

   4. Applied egg-rr 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/r* 86.2%

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

      5. unpow2 86.2%

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

      6. rem-sqrt-square 86.2%

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

      7. unpow2 86.2%

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

      8. rem-sqrt-square 97.9%

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

   6. Simplified 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. add-sqr-sqrt 60.3%

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

      3. fabs-sqr 60.3%

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

      4. add-sqr-sqrt 66.9%

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

      5. times-frac 67.0%

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

      6. *-commutative 67.0%

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

   8. Applied egg-rr 67.0%

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

   if 1.20000000000000001e-27 < x

   1. Initial program 54.8%

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

      1. *-un-lft-identity 54.8%

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

      2. add-sqr-sqrt 54.8%

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

      3. times-frac 54.8%

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

      4. sqrt-prod 54.8%

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

      5. sqrt-pow1 44.2%

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

      6. metadata-eval 44.2%

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

      7. pow1 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 42.6%

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

      10. unpow2 42.6%

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

      11. pow-prod-down 44.2%

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

      12. sqrt-prod 44.2%

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

   4. Applied egg-rr 80.9%

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

      1. associate-*l/ 81.0%

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

      2. *-lft-identity 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 81.1%

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

      5. unpow2 81.1%

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

      6. rem-sqrt-square 81.1%

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

      7. unpow2 81.1%

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

      8. rem-sqrt-square 96.7%

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

   6. Simplified 96.7%

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

      1. div-inv 96.7%

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

      2. times-frac 96.7%

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

      3. associate-/l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. add-sqr-sqrt 51.9%

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

      6. fabs-sqr 51.9%

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

      7. add-sqr-sqrt 74.7%

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

      8. add-sqr-sqrt 52.0%

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

      9. fabs-sqr 52.0%

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

      10. add-sqr-sqrt 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. *-commutative 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. frac-times 96.6%

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

      4. *-commutative 96.6%

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

      5. un-div-inv 96.7%

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

      6. times-frac 96.7%

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

      7. *-commutative 96.7%

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

      8. frac-times 96.6%

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

      9. *-rgt-identity 96.6%

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

      10. associate-*l* 93.1%

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

      11. associate-*l* 95.9%

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

   10. Applied egg-rr 95.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{x}}{c}}{c \cdot \left|x \cdot s\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(\frac{t\_1}{s\_m} \cdot \frac{1}{x\_m}\right) \cdot \frac{1}{c\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 1.8e-27)
     (/ (* (* (/ t_1 s_m) (/ 1.0 x_m)) (/ 1.0 c_m)) (* c_m (* x_m s_m)))
     (/ (/ t_1 t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.8e-27) {
		tmp = (((t_1 / s_m) * (1.0 / x_m)) * (1.0 / c_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 1.8d-27) then
        tmp = (((t_1 / s_m) * (1.0d0 / x_m)) * (1.0d0 / c_m)) / (c_m * (x_m * s_m))
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1.8e-27) {
		tmp = (((t_1 / s_m) * (1.0 / x_m)) * (1.0 / c_m)) / (c_m * (x_m * s_m));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 1.8e-27:
		tmp = (((t_1 / s_m) * (1.0 / x_m)) * (1.0 / c_m)) / (c_m * (x_m * s_m))
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 1.8e-27)
		tmp = Float64(Float64(Float64(Float64(t_1 / s_m) * Float64(1.0 / x_m)) * Float64(1.0 / c_m)) / Float64(c_m * Float64(x_m * s_m)));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 1.8e-27)
		tmp = (((t_1 / s_m) * (1.0 / x_m)) * (1.0 / c_m)) / (c_m * (x_m * s_m));
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 1.8e-27], N[(N[(N[(N[(t$95$1 / s$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7999999999999999e-27

   1. Initial program 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. add-sqr-sqrt 67.1%

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

      3. times-frac 67.1%

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

      4. sqrt-prod 67.1%

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

      5. sqrt-pow1 45.6%

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

      6. metadata-eval 45.6%

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

      7. pow1 45.6%

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

      8. *-commutative 45.6%

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

      9. associate-*r* 42.8%

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

      10. unpow2 42.8%

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

      11. pow-prod-down 45.6%

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

      12. sqrt-prod 45.6%

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

   4. Applied egg-rr 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/r* 86.2%

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

      5. unpow2 86.2%

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

      6. rem-sqrt-square 86.2%

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

      7. unpow2 86.2%

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

      8. rem-sqrt-square 97.9%

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

   6. Simplified 97.9%

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

      1. div-inv 97.8%

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

      2. *-commutative 97.8%

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

      3. add-sqr-sqrt 60.2%

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

      4. fabs-sqr 60.2%

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

      5. add-sqr-sqrt 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. associate-/r* 66.9%

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

      2. div-inv 66.9%

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

   10. Applied egg-rr 66.9%

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

       1. pow1 66.9%

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

       2. pow1 66.9%

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

       3. metadata-eval 66.9%

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

       4. sqrt-pow1 60.5%

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

       5. sqrt-pow1 66.9%

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

       6. metadata-eval 66.9%

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

       7. pow1 66.9%

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

       8. add-sqr-sqrt 52.8%

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

       9. fabs-sqr 52.8%

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

       10. add-sqr-sqrt 97.9%

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

   12. Applied egg-rr 97.9%

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

       1. unpow1 97.9%

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

   14. Simplified 97.9%

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

   if 1.7999999999999999e-27 < x

   1. Initial program 54.8%

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

      1. *-un-lft-identity 54.8%

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

      2. add-sqr-sqrt 54.8%

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

      3. times-frac 54.8%

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

      4. sqrt-prod 54.8%

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

      5. sqrt-pow1 44.2%

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

      6. metadata-eval 44.2%

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

      7. pow1 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 42.6%

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

      10. unpow2 42.6%

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

      11. pow-prod-down 44.2%

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

      12. sqrt-prod 44.2%

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

   4. Applied egg-rr 80.9%

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

      1. associate-*l/ 81.0%

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

      2. *-lft-identity 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 81.1%

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

      5. unpow2 81.1%

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

      6. rem-sqrt-square 81.1%

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

      7. unpow2 81.1%

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

      8. rem-sqrt-square 96.7%

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

   6. Simplified 96.7%

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

      1. div-inv 96.7%

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

      2. times-frac 96.7%

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

      3. associate-/l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. add-sqr-sqrt 51.9%

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

      6. fabs-sqr 51.9%

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

      7. add-sqr-sqrt 74.7%

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

      8. add-sqr-sqrt 52.0%

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

      9. fabs-sqr 52.0%

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

      10. add-sqr-sqrt 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. *-commutative 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. frac-times 96.6%

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

      4. *-commutative 96.6%

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

      5. un-div-inv 96.7%

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

      6. times-frac 96.7%

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

      7. *-commutative 96.7%

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

      8. frac-times 96.6%

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

      9. *-rgt-identity 96.6%

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

      10. associate-*l* 93.1%

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

      11. associate-*l* 95.9%

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

   10. Applied egg-rr 95.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(\frac{\cos \left(x \cdot 2\right)}{s} \cdot \frac{1}{x}\right) \cdot \frac{1}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 1.5e-27)
     (pow (* c_m (* x_m s_m)) -2.0)
     (/ (/ (cos (* x_m 2.0)) t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 1.5e-27) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 1.5d-27) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 1.5e-27) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 1.5e-27:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (math.cos((x_m * 2.0)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 1.5e-27)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 1.5e-27)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.5e-27], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e-27

   1. Initial program 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. add-sqr-sqrt 67.1%

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

      3. times-frac 67.1%

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

      4. sqrt-prod 67.1%

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

      5. sqrt-pow1 45.6%

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

      6. metadata-eval 45.6%

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

      7. pow1 45.6%

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

      8. *-commutative 45.6%

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

      9. associate-*r* 42.8%

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

      10. unpow2 42.8%

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

      11. pow-prod-down 45.6%

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

      12. sqrt-prod 45.6%

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

   4. Applied egg-rr 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-/r* 86.2%

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

      5. unpow2 86.2%

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

      6. rem-sqrt-square 86.2%

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

      7. unpow2 86.2%

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

      8. rem-sqrt-square 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in x around 0 70.3%

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

      1. associate-/r* 70.3%

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

      2. unpow2 70.3%

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

      3. sqr-abs 70.3%

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

      4. unpow2 70.3%

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

      5. associate-/r* 70.3%

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

      6. unpow2 70.3%

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

      7. unpow2 70.3%

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

      8. swap-sqr 86.3%

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

      9. unpow2 86.3%

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

      10. unpow-1 86.3%

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

      11. exp-to-pow 54.2%

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

      12. *-commutative 54.2%

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

      13. exp-prod 54.5%

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

      14. *-commutative 54.5%

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

      15. associate-*r* 54.5%

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

      16. metadata-eval 54.5%

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

      17. *-commutative 54.5%

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

      18. exp-to-pow 86.9%

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

   9. Simplified 86.9%

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

   if 1.5000000000000001e-27 < x

   1. Initial program 54.8%

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

      1. *-un-lft-identity 54.8%

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

      2. add-sqr-sqrt 54.8%

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

      3. times-frac 54.8%

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

      4. sqrt-prod 54.8%

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

      5. sqrt-pow1 44.2%

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

      6. metadata-eval 44.2%

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

      7. pow1 44.2%

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

      8. *-commutative 44.2%

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

      9. associate-*r* 42.6%

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

      10. unpow2 42.6%

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

      11. pow-prod-down 44.2%

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

      12. sqrt-prod 44.2%

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

   4. Applied egg-rr 80.9%

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

      1. associate-*l/ 81.0%

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

      2. *-lft-identity 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 81.1%

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

      5. unpow2 81.1%

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

      6. rem-sqrt-square 81.1%

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

      7. unpow2 81.1%

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

      8. rem-sqrt-square 96.7%

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

   6. Simplified 96.7%

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

      1. div-inv 96.7%

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

      2. times-frac 96.7%

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

      3. associate-/l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. add-sqr-sqrt 51.9%

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

      6. fabs-sqr 51.9%

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

      7. add-sqr-sqrt 74.7%

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

      8. add-sqr-sqrt 52.0%

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

      9. fabs-sqr 52.0%

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

      10. add-sqr-sqrt 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. *-commutative 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. frac-times 96.6%

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

      4. *-commutative 96.6%

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

      5. un-div-inv 96.7%

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

      6. times-frac 96.7%

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

      7. *-commutative 96.7%

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

      8. frac-times 96.6%

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

      9. *-rgt-identity 96.6%

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

      10. associate-*l* 93.1%

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

      11. associate-*l* 95.9%

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

   10. Applied egg-rr 95.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 2.7× speedup

Math

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

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 (* c_m (* x_m s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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(c_m * Float64(x_m * s_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 = c_m * (x_m * s_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[(c$95$m * N[(x$95$m * s$95$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 64.0%

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

   1. associate-/l/ 64.0%

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

   2. remove-double-neg 64.0%

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

   3. distribute-frac-neg 64.0%

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

   4. distribute-neg-frac 64.0%

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

   5. remove-double-neg 64.0%

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

   6. *-commutative 64.0%

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

   7. associate-*r* 60.7%

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

   8. unpow2 60.7%

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

   9. associate-/r* 60.3%

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

   10. cos-neg 60.3%

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

   11. *-commutative 60.3%

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

   12. distribute-rgt-neg-in 60.3%

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

   13. metadata-eval 60.3%

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

3. Simplified 60.3%

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

5. Taylor expanded in x around inf 60.7%

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

   1. associate-/r* 60.7%

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

   2. unpow2 60.7%

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

   3. unpow2 60.7%

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

   4. swap-sqr 76.2%

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

   5. unpow2 76.2%

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

   6. associate-/r* 76.4%

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

   7. *-commutative 76.4%

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

   8. unpow2 76.4%

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

   9. rem-square-sqrt 76.4%

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

   10. swap-sqr 84.5%

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

   11. unpow2 84.5%

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

   12. unpow2 84.5%

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

   13. rem-sqrt-square 97.0%

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

7. Simplified 97.0%

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

   1. pow1 78.3%

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

   2. metadata-eval 78.3%

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

   3. sqrt-pow1 78.3%

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

   4. pow2 78.3%

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

   5. sqrt-pow1 58.7%

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

   6. metadata-eval 58.7%

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

   7. pow1 58.7%

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

   8. add-sqr-sqrt 32.9%

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

   9. fabs-sqr 32.9%

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

   10. add-sqr-sqrt 56.1%

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

   11. sqrt-pow1 59.2%

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

   12. metadata-eval 59.2%

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

   13. pow1 59.2%

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

   14. add-sqr-sqrt 41.0%

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

   15. fabs-sqr 41.0%

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

   16. add-sqr-sqrt 78.3%

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

9. Applied egg-rr 97.0%

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

10. Final simplification 97.0%

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

Alternative 5: 79.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

   1. *-un-lft-identity 64.0%

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

   2. add-sqr-sqrt 64.0%

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

   3. times-frac 63.9%

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

   4. sqrt-prod 64.0%

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

   5. sqrt-pow1 45.2%

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

   6. metadata-eval 45.2%

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

   7. pow1 45.2%

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

   8. *-commutative 45.2%

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

   9. associate-*r* 42.7%

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

   10. unpow2 42.7%

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

   11. pow-prod-down 45.2%

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

   12. sqrt-prod 45.2%

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

4. Applied egg-rr 84.9%

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

   1. associate-*l/ 84.9%

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

   2. *-lft-identity 84.9%

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

   3. *-commutative 84.9%

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

   4. associate-/r* 84.9%

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

   5. unpow2 84.9%

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

   6. rem-sqrt-square 84.9%

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

   7. unpow2 84.9%

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

   8. rem-sqrt-square 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in x around 0 64.7%

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

   1. associate-/r* 64.7%

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

   2. unpow2 64.7%

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

   3. sqr-abs 64.7%

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

   4. unpow2 64.7%

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

   5. associate-/r* 64.7%

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

   6. unpow2 64.7%

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

   7. unpow2 64.7%

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

   8. swap-sqr 78.3%

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

   9. unpow2 78.3%

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

   10. unpow-1 78.3%

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

   11. exp-to-pow 46.3%

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

   12. *-commutative 46.3%

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

   13. exp-prod 46.5%

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

   14. *-commutative 46.5%

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

   15. associate-*r* 46.5%

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

   16. metadata-eval 46.5%

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

   17. *-commutative 46.5%

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

   18. exp-to-pow 78.7%

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

9. Simplified 78.7%

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

10. Final simplification 78.7%

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

Alternative 6: 79.6% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in x around 0 55.1%

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

   1. associate-/r* 55.0%

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

   2. unpow2 55.0%

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

   3. unpow2 55.0%

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

   4. swap-sqr 64.7%

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

   5. unpow2 64.7%

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

   6. associate-/r* 64.7%

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

   7. unpow2 64.7%

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

   8. rem-square-sqrt 64.7%

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

   9. swap-sqr 71.0%

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

   10. unpow2 71.0%

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

   11. unpow2 71.0%

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

   12. rem-sqrt-square 78.3%

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

5. Simplified 78.3%

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

   1. pow1 78.3%

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

   2. metadata-eval 78.3%

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

   3. sqrt-pow1 78.3%

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

   4. pow2 78.3%

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

   5. add-sqr-sqrt 78.3%

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

   6. inv-pow 78.3%

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

   7. unpow2 78.3%

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

   8. unpow-prod-down 78.7%

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

   9. inv-pow 78.7%

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

   10. add-sqr-sqrt 46.7%

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

   11. fabs-sqr 46.7%

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

   12. add-sqr-sqrt 59.3%

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

   13. inv-pow 59.3%

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

   14. add-sqr-sqrt 41.2%

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

   15. fabs-sqr 41.2%

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

   16. add-sqr-sqrt 78.7%

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

7. Applied egg-rr 78.7%

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

8. Final simplification 78.7%

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

Alternative 7: 79.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in x around 0 55.1%

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

   1. associate-/r* 55.0%

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

   2. unpow2 55.0%

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

   3. unpow2 55.0%

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

   4. swap-sqr 64.7%

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

   5. unpow2 64.7%

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

   6. associate-/r* 64.7%

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

   7. unpow2 64.7%

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

   8. rem-square-sqrt 64.7%

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

   9. swap-sqr 71.0%

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

   10. unpow2 71.0%

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

   11. unpow2 71.0%

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

   12. rem-sqrt-square 78.3%

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

5. Simplified 78.3%

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

   1. pow1 78.3%

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

   2. metadata-eval 78.3%

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

   3. sqrt-pow1 78.3%

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

   4. pow2 78.3%

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

   5. sqrt-pow1 58.7%

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

   6. metadata-eval 58.7%

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

   7. pow1 58.7%

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

   8. add-sqr-sqrt 32.9%

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

   9. fabs-sqr 32.9%

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

   10. add-sqr-sqrt 56.1%

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

   11. sqrt-pow1 59.2%

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

   12. metadata-eval 59.2%

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

   13. pow1 59.2%

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

   14. add-sqr-sqrt 41.0%

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

   15. fabs-sqr 41.0%

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

   16. add-sqr-sqrt 78.3%

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

7. Applied egg-rr 78.3%

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

8. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2024067 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))