mixedcos

Percentage Accurate: 67.3% → 99.6%

Time: 12.5s

Alternatives: 11

Speedup: 24.1×

• 30.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot -2\right)\\ \mathbf{if}\;x\_m \leq 50:\\ \;\;\;\;t\_1 \cdot {\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))) (t_1 (cos (* x_m -2.0))))
   (if (<= x_m 50.0)
     (* t_1 (pow (* c_m (* x_m s_m)) -2.0))
     (/ (* t_1 (/ 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 = s_m * (x_m * c_m);
	double t_1 = cos((x_m * -2.0));
	double tmp;
	if (x_m <= 50.0) {
		tmp = t_1 * pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (t_1 * (1.0 / t_0)) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * (-2.0d0)))
    if (x_m <= 50.0d0) then
        tmp = t_1 * ((c_m * (x_m * s_m)) ** (-2.0d0))
    else
        tmp = (t_1 * (1.0d0 / t_0)) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = Math.cos((x_m * -2.0));
	double tmp;
	if (x_m <= 50.0) {
		tmp = t_1 * Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (t_1 * (1.0 / t_0)) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	t_1 = math.cos((x_m * -2.0))
	tmp = 0
	if x_m <= 50.0:
		tmp = t_1 * math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (t_1 * (1.0 / t_0)) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	t_1 = cos(Float64(x_m * -2.0))
	tmp = 0.0
	if (x_m <= 50.0)
		tmp = Float64(t_1 * (Float64(c_m * Float64(x_m * s_m)) ^ -2.0));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / t_0)) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	t_1 = cos((x_m * -2.0));
	tmp = 0.0;
	if (x_m <= 50.0)
		tmp = t_1 * ((c_m * (x_m * s_m)) ^ -2.0);
	else
		tmp = (t_1 * (1.0 / t_0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 50.0], N[(t$95$1 * N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 50

   1. Initial program 67.9%

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

      1. associate-/r* 67.9%

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

      2. cos-neg 67.9%

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

      3. distribute-rgt-neg-out 67.9%

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

      4. distribute-rgt-neg-out 67.9%

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

      5. *-commutative 67.9%

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

      6. distribute-rgt-neg-in 67.9%

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

      7. metadata-eval 67.9%

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

      8. *-commutative 67.9%

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

      9. associate-*l* 61.5%

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

      10. unpow2 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around inf 61.5%

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

      1. associate-/r* 61.5%

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

      2. *-commutative 61.5%

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

      3. *-commutative 61.5%

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

      4. unpow2 61.5%

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

      5. unpow2 61.5%

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

      6. swap-sqr 77.1%

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

      7. unpow2 77.1%

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

      8. associate-/l/ 77.4%

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

      9. unpow2 77.4%

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

      10. unpow2 77.4%

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

      11. swap-sqr 96.7%

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

      12. unpow2 96.7%

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

      13. *-commutative 96.7%

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

      14. associate-*l* 97.4%

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

   7. Simplified 97.4%

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

      1. div-inv 97.4%

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

      2. pow-flip 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 95.6%

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

      5. metadata-eval 95.6%

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

   9. Applied egg-rr 95.6%

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

       1. *-commutative 95.6%

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

       2. associate-*r* 97.6%

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

   11. Simplified 97.6%

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

   if 50 < x

   1. Initial program 57.1%

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

      1. associate-/r* 56.7%

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

      2. cos-neg 56.7%

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

      3. distribute-rgt-neg-out 56.7%

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

      4. distribute-rgt-neg-out 56.7%

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

      5. *-commutative 56.7%

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

      6. distribute-rgt-neg-in 56.7%

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

      7. metadata-eval 56.7%

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

      8. *-commutative 56.7%

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

      9. associate-*l* 50.6%

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

      10. unpow2 50.6%

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

   3. Simplified 50.6%

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

      1. div-inv 50.5%

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

      2. add-sqr-sqrt 50.5%

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

      3. times-frac 50.5%

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

      4. pow-prod-down 50.6%

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

      5. sqrt-pow1 44.4%

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

      6. metadata-eval 44.4%

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

      7. pow1 44.4%

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

      8. *-commutative 44.4%

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

      9. pow-flip 44.3%

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

      10. metadata-eval 44.3%

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

      11. pow-prod-down 51.6%

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

      12. sqrt-pow1 76.5%

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

      13. metadata-eval 76.5%

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

      14. pow1 76.5%

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

      15. *-commutative 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. sqr-pow 76.4%

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

      2. associate-/l* 84.9%

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

      3. metadata-eval 84.9%

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

      4. unpow-1 85.0%

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

      5. metadata-eval 85.0%

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

      6. unpow-1 85.0%

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

   8. Applied egg-rr 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*l/ 85.0%

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

      3. *-un-lft-identity 85.0%

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

      4. frac-times 91.5%

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

      5. associate-/l/ 91.4%

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

      6. *-commutative 91.4%

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

      7. associate-*r* 87.1%

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

      8. associate-*r* 95.1%

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

      9. *-commutative 95.1%

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

      10. *-commutative 95.1%

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

   10. Applied egg-rr 95.1%

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

4. Final simplification 96.9%

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

Alternative 2: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot -2\right)\\ t_1 := x\_m \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;{c\_m}^{2} \leq 0:\\ \;\;\;\;\frac{1}{t\_1} \cdot \frac{t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot s\_m} \cdot \frac{\frac{1}{c\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m -2.0))) (t_1 (* x_m (* c_m s_m))))
   (if (<= (pow c_m 2.0) 0.0)
     (* (/ 1.0 t_1) (/ t_0 t_1))
     (* (/ t_0 (* x_m s_m)) (/ (/ 1.0 c_m) (* c_m (* x_m s_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * -2.0));
	double t_1 = x_m * (c_m * s_m);
	double tmp;
	if (pow(c_m, 2.0) <= 0.0) {
		tmp = (1.0 / t_1) * (t_0 / t_1);
	} else {
		tmp = (t_0 / (x_m * s_m)) * ((1.0 / c_m) / (c_m * (x_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * (-2.0d0)))
    t_1 = x_m * (c_m * s_m)
    if ((c_m ** 2.0d0) <= 0.0d0) then
        tmp = (1.0d0 / t_1) * (t_0 / t_1)
    else
        tmp = (t_0 / (x_m * s_m)) * ((1.0d0 / c_m) / (c_m * (x_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * -2.0));
	double t_1 = x_m * (c_m * s_m);
	double tmp;
	if (Math.pow(c_m, 2.0) <= 0.0) {
		tmp = (1.0 / t_1) * (t_0 / t_1);
	} else {
		tmp = (t_0 / (x_m * s_m)) * ((1.0 / c_m) / (c_m * (x_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * -2.0))
	t_1 = x_m * (c_m * s_m)
	tmp = 0
	if math.pow(c_m, 2.0) <= 0.0:
		tmp = (1.0 / t_1) * (t_0 / t_1)
	else:
		tmp = (t_0 / (x_m * s_m)) * ((1.0 / c_m) / (c_m * (x_m * s_m)))
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[c$95$m, 2.0], $MachinePrecision], 0.0], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 0.0

   1. Initial program 57.3%

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

      1. associate-/r* 57.3%

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

      2. cos-neg 57.3%

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

      3. distribute-rgt-neg-out 57.3%

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

      4. distribute-rgt-neg-out 57.3%

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

      5. *-commutative 57.3%

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

      6. distribute-rgt-neg-in 57.3%

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

      7. metadata-eval 57.3%

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

      8. *-commutative 57.3%

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

      9. associate-*l* 51.0%

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

      10. unpow2 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in x around inf 51.0%

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

      1. associate-/r* 51.0%

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

      2. *-commutative 51.0%

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

      3. *-commutative 51.0%

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

      4. unpow2 51.0%

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

      5. unpow2 51.0%

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

      6. swap-sqr 69.3%

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

      7. unpow2 69.3%

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

      8. associate-/l/ 69.3%

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

      9. unpow2 69.3%

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

      10. unpow2 69.3%

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

      11. swap-sqr 90.5%

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

      12. unpow2 90.5%

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

      13. *-commutative 90.5%

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

      14. associate-*l* 97.4%

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

   7. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. unpow2 97.4%

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

      3. times-frac 98.4%

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

      4. *-commutative 98.4%

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

      5. associate-*l* 97.4%

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

      6. *-commutative 97.4%

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

      7. associate-*l* 98.7%

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

   9. Applied egg-rr 98.7%

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

   if 0.0 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 67.7%

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

      1. associate-/r* 67.6%

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

      2. cos-neg 67.6%

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

      3. distribute-rgt-neg-out 67.6%

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

      4. distribute-rgt-neg-out 67.6%

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

      5. *-commutative 67.6%

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

      6. distribute-rgt-neg-in 67.6%

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

      7. metadata-eval 67.6%

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

      8. *-commutative 67.6%

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

      9. associate-*l* 61.3%

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

      10. unpow2 61.3%

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

   3. Simplified 61.3%

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

      1. div-inv 61.3%

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

      2. add-sqr-sqrt 61.2%

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

      3. times-frac 61.2%

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

      4. pow-prod-down 61.2%

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

      5. sqrt-pow1 49.0%

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

      6. metadata-eval 49.0%

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

      7. pow1 49.0%

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

      8. *-commutative 49.0%

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

      9. pow-flip 49.0%

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

      10. metadata-eval 49.0%

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

      11. pow-prod-down 57.9%

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

      12. sqrt-pow1 83.7%

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

      13. metadata-eval 83.7%

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

      14. pow1 83.7%

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

      15. *-commutative 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. sqr-pow 83.6%

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

      2. times-frac 91.4%

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

      3. metadata-eval 91.4%

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

      4. unpow-1 91.4%

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

      5. metadata-eval 91.4%

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

      6. unpow-1 91.5%

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

   8. Applied egg-rr 91.5%

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

      1. associate-/l/ 91.4%

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

      2. frac-times 92.7%

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

      3. *-un-lft-identity 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*l* 95.0%

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

   10. Applied egg-rr 95.0%

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

Alternative 3: 97.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot -2\right)\\ \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{elif}\;x\_m \leq 5.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m \cdot c\_m}}{s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m -2.0))))
   (if (<= x_m 2.8e-16)
     (pow (* c_m (* x_m s_m)) -2.0)
     (if (<= x_m 5.4e+130)
       (/ (/ t_0 x_m) (* s_m (* (* x_m c_m) (* c_m s_m))))
       (/ (/ t_0 (* x_m c_m)) (* s_m (* s_m (* x_m c_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * -2.0));
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else if (x_m <= 5.4e+130) {
		tmp = (t_0 / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	} else {
		tmp = (t_0 / (x_m * c_m)) / (s_m * (s_m * (x_m * c_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x_m * (-2.0d0)))
    if (x_m <= 2.8d-16) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else if (x_m <= 5.4d+130) then
        tmp = (t_0 / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)))
    else
        tmp = (t_0 / (x_m * c_m)) / (s_m * (s_m * (x_m * c_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * -2.0));
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else if (x_m <= 5.4e+130) {
		tmp = (t_0 / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	} else {
		tmp = (t_0 / (x_m * c_m)) / (s_m * (s_m * (x_m * c_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * -2.0))
	tmp = 0
	if x_m <= 2.8e-16:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	elif x_m <= 5.4e+130:
		tmp = (t_0 / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)))
	else:
		tmp = (t_0 / (x_m * c_m)) / (s_m * (s_m * (x_m * c_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m * -2.0))
	tmp = 0.0
	if (x_m <= 2.8e-16)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	elseif (x_m <= 5.4e+130)
		tmp = Float64(Float64(t_0 / x_m) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(c_m * s_m))));
	else
		tmp = Float64(Float64(t_0 / Float64(x_m * c_m)) / Float64(s_m * Float64(s_m * Float64(x_m * c_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * -2.0));
	tmp = 0.0;
	if (x_m <= 2.8e-16)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	elseif (x_m <= 5.4e+130)
		tmp = (t_0 / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	else
		tmp = (t_0 / (x_m * c_m)) / (s_m * (s_m * (x_m * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 2.8e-16], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x$95$m, 5.4e+130], N[(N[(t$95$0 / x$95$m), $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.8000000000000001e-16

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 60.5%

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

      10. unpow2 60.5%

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

   3. Simplified 60.5%

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

      1. div-inv 60.5%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 60.4%

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

      4. pow-prod-down 60.4%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. *-commutative 42.4%

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

      9. pow-flip 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow-prod-down 52.3%

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

      12. sqrt-pow1 80.8%

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

      13. metadata-eval 80.8%

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

      14. pow1 80.8%

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

      15. *-commutative 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 71.9%

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

   8. Taylor expanded in x around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. *-commutative 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. swap-sqr 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*r* 85.9%

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

      12. unpow2 85.9%

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

      13. exp-to-pow 55.1%

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

      14. *-commutative 55.1%

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

      15. exp-neg 55.2%

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

      16. *-commutative 55.2%

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

      17. distribute-rgt-neg-in 55.2%

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

      18. metadata-eval 55.2%

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

      19. exp-to-pow 86.5%

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

   10. Simplified 85.3%

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

   if 2.8000000000000001e-16 < x < 5.3999999999999997e130

   1. Initial program 55.7%

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

      1. associate-/r* 55.7%

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

      2. cos-neg 55.7%

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

      3. distribute-rgt-neg-out 55.7%

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

      4. distribute-rgt-neg-out 55.7%

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

      5. *-commutative 55.7%

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

      6. distribute-rgt-neg-in 55.7%

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

      7. metadata-eval 55.7%

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

      8. *-commutative 55.7%

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

      9. associate-*l* 55.8%

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

      10. unpow2 55.8%

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

   3. Simplified 55.8%

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

      1. div-inv 55.7%

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

      2. add-sqr-sqrt 55.7%

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

      3. times-frac 55.7%

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

      4. pow-prod-down 55.8%

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

      5. sqrt-pow1 37.8%

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

      6. metadata-eval 37.8%

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

      7. pow1 37.8%

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

      8. *-commutative 37.8%

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

      9. pow-flip 37.8%

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

      10. metadata-eval 37.8%

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

      11. pow-prod-down 37.8%

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

      12. sqrt-pow1 71.6%

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

      13. metadata-eval 71.6%

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

      14. pow1 71.6%

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

      15. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. sqr-pow 71.7%

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

      2. associate-/l* 78.3%

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

      3. metadata-eval 78.3%

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

      4. unpow-1 78.4%

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

      5. metadata-eval 78.4%

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

      6. unpow-1 78.4%

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

   8. Applied egg-rr 78.4%

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

   10. Applied egg-rr 86.4%

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

   if 5.3999999999999997e130 < x

   1. Initial program 63.9%

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

      1. associate-/r* 63.2%

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

      2. cos-neg 63.2%

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

      3. distribute-rgt-neg-out 63.2%

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

      4. distribute-rgt-neg-out 63.2%

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

      5. *-commutative 63.2%

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

      6. distribute-rgt-neg-in 63.2%

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

      7. metadata-eval 63.2%

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

      8. *-commutative 63.2%

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

      9. associate-*l* 52.2%

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

      10. unpow2 52.2%

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

   3. Simplified 52.2%

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

      1. div-inv 52.2%

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

      2. add-sqr-sqrt 52.2%

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

      3. times-frac 52.2%

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

      4. pow-prod-down 52.2%

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

      5. sqrt-pow1 52.2%

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

      6. metadata-eval 52.2%

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

      7. pow1 52.2%

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

      8. *-commutative 52.2%

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

      9. pow-flip 52.2%

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

      10. metadata-eval 52.2%

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

      11. pow-prod-down 65.0%

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

      12. sqrt-pow1 83.8%

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

      13. metadata-eval 83.8%

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

      14. pow1 83.8%

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

      15. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. sqr-pow 83.7%

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

      2. associate-/l* 92.8%

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

      3. metadata-eval 92.8%

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

      4. unpow-1 92.8%

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

      5. metadata-eval 92.8%

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

      6. unpow-1 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-/r* 92.8%

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

      2. frac-times 92.8%

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

      3. *-un-lft-identity 92.8%

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

      4. frac-times 89.6%

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

      5. div-inv 89.6%

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

      6. associate-*r* 89.6%

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

      7. div-inv 89.6%

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

      8. associate-/l/ 89.6%

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

      9. *-commutative 89.6%

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

      10. un-div-inv 89.6%

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

      11. associate-*r* 89.6%

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

      12. *-commutative 89.6%

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

      13. *-commutative 89.6%

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

   10. Applied egg-rr 89.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{x}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{x \cdot c}}{s \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right) \cdot \frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 2.8e-16)
     (pow (* c_m (* x_m s_m)) -2.0)
     (/ (* (cos (* x_m -2.0)) (/ 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 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (cos((x_m * -2.0)) * (1.0 / 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 <= 2.8d-16) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = (cos((x_m * (-2.0d0))) * (1.0d0 / 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 <= 2.8e-16) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * -2.0)) * (1.0 / 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 <= 2.8e-16:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (math.cos((x_m * -2.0)) * (1.0 / 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 <= 2.8e-16)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * -2.0)) * Float64(1.0 / 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 <= 2.8e-16)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (cos((x_m * -2.0)) * (1.0 / 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, 2.8e-16], 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] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e-16

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 60.5%

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

      10. unpow2 60.5%

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

   3. Simplified 60.5%

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

      1. div-inv 60.5%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 60.4%

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

      4. pow-prod-down 60.4%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. *-commutative 42.4%

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

      9. pow-flip 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow-prod-down 52.3%

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

      12. sqrt-pow1 80.8%

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

      13. metadata-eval 80.8%

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

      14. pow1 80.8%

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

      15. *-commutative 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 71.9%

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

   8. Taylor expanded in x around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. *-commutative 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. swap-sqr 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*r* 85.9%

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

      12. unpow2 85.9%

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

      13. exp-to-pow 55.1%

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

      14. *-commutative 55.1%

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

      15. exp-neg 55.2%

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

      16. *-commutative 55.2%

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

      17. distribute-rgt-neg-in 55.2%

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

      18. metadata-eval 55.2%

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

      19. exp-to-pow 86.5%

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

   10. Simplified 85.3%

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

   if 2.8000000000000001e-16 < x

   1. Initial program 60.0%

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

      1. associate-/r* 59.6%

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

      2. cos-neg 59.6%

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

      3. distribute-rgt-neg-out 59.6%

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

      4. distribute-rgt-neg-out 59.6%

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

      5. *-commutative 59.6%

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

      6. distribute-rgt-neg-in 59.6%

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

      7. metadata-eval 59.6%

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

      8. *-commutative 59.6%

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

      9. associate-*l* 53.9%

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

      10. unpow2 53.9%

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

   3. Simplified 53.9%

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

      1. div-inv 53.9%

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

      2. add-sqr-sqrt 53.9%

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

      3. times-frac 53.9%

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

      4. pow-prod-down 53.9%

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

      5. sqrt-pow1 45.4%

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

      6. metadata-eval 45.4%

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

      7. pow1 45.4%

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

      8. *-commutative 45.4%

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

      9. pow-flip 45.4%

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

      10. metadata-eval 45.4%

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

      11. pow-prod-down 52.2%

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

      12. sqrt-pow1 78.0%

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

      13. metadata-eval 78.0%

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

      14. pow1 78.0%

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

      15. *-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. sqr-pow 78.0%

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

      2. associate-/l* 86.0%

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

      3. metadata-eval 86.0%

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

      4. unpow-1 86.0%

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

      5. metadata-eval 86.0%

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

      6. unpow-1 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*l/ 86.0%

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

      3. *-un-lft-identity 86.0%

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

      4. frac-times 92.1%

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

      5. associate-/l/ 91.9%

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

      6. *-commutative 91.9%

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

      7. associate-*r* 88.0%

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

      8. associate-*r* 95.5%

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

      9. *-commutative 95.5%

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

      10. *-commutative 95.5%

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

   10. Applied egg-rr 95.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right) \cdot \frac{1}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{\cos \left(x\_m \cdot -2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* x_m (* c_m s_m))))
   (if (<= x_m 2.5e-16)
     (pow (* c_m (* x_m s_m)) -2.0)
     (* (/ 1.0 t_0) (/ (cos (* x_m -2.0)) t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (c_m * s_m);
	double tmp;
	if (x_m <= 2.5e-16) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (1.0 / t_0) * (cos((x_m * -2.0)) / 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 = x_m * (c_m * s_m)
    if (x_m <= 2.5d-16) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = (1.0d0 / t_0) * (cos((x_m * (-2.0d0))) / t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (c_m * s_m);
	double tmp;
	if (x_m <= 2.5e-16) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (1.0 / t_0) * (Math.cos((x_m * -2.0)) / t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (c_m * s_m)
	tmp = 0
	if x_m <= 2.5e-16:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (1.0 / t_0) * (math.cos((x_m * -2.0)) / 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(x_m * Float64(c_m * s_m))
	tmp = 0.0
	if (x_m <= 2.5e-16)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x_m * -2.0)) / t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = x_m * (c_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.5e-16)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (1.0 / t_0) * (cos((x_m * -2.0)) / 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[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-16], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000002e-16

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 60.5%

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

      10. unpow2 60.5%

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

   3. Simplified 60.5%

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

      1. div-inv 60.5%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 60.4%

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

      4. pow-prod-down 60.4%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. *-commutative 42.4%

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

      9. pow-flip 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow-prod-down 52.3%

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

      12. sqrt-pow1 80.8%

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

      13. metadata-eval 80.8%

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

      14. pow1 80.8%

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

      15. *-commutative 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 71.9%

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

   8. Taylor expanded in x around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. *-commutative 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. swap-sqr 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*r* 85.9%

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

      12. unpow2 85.9%

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

      13. exp-to-pow 55.1%

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

      14. *-commutative 55.1%

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

      15. exp-neg 55.2%

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

      16. *-commutative 55.2%

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

      17. distribute-rgt-neg-in 55.2%

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

      18. metadata-eval 55.2%

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

      19. exp-to-pow 86.5%

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

   10. Simplified 85.3%

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

   if 2.5000000000000002e-16 < x

   1. Initial program 60.0%

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

      1. associate-/r* 59.6%

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

      2. cos-neg 59.6%

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

      3. distribute-rgt-neg-out 59.6%

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

      4. distribute-rgt-neg-out 59.6%

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

      5. *-commutative 59.6%

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

      6. distribute-rgt-neg-in 59.6%

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

      7. metadata-eval 59.6%

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

      8. *-commutative 59.6%

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

      9. associate-*l* 53.9%

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

      10. unpow2 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in x around inf 54.0%

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

      1. associate-/r* 53.9%

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

      2. *-commutative 53.9%

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

      3. *-commutative 53.9%

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

      4. unpow2 53.9%

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

      5. unpow2 53.9%

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

      6. swap-sqr 67.8%

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

      7. unpow2 67.8%

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

      8. associate-/l/ 68.2%

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

      9. unpow2 68.2%

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

      10. unpow2 68.2%

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

      11. swap-sqr 91.7%

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

      12. unpow2 91.7%

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

      13. *-commutative 91.7%

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

      14. associate-*l* 95.4%

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

   7. Simplified 95.4%

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

      1. *-un-lft-identity 95.4%

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

      2. unpow2 95.4%

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

      3. times-frac 95.5%

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

      4. *-commutative 95.5%

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

      5. associate-*l* 93.3%

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

      6. *-commutative 93.3%

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

      7. associate-*l* 97.3%

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

   9. Applied egg-rr 97.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(c \cdot s\right)} \cdot \frac{\cos \left(x \cdot -2\right)}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot -2\right)}{x\_m}}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 2.8e-16)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (/ (cos (* x_m -2.0)) x_m) (* s_m (* (* x_m c_m) (* c_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (cos((x_m * -2.0)) / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.8d-16) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = (cos((x_m * (-2.0d0))) / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * -2.0)) / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2.8e-16:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (math.cos((x_m * -2.0)) / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2.8e-16)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * -2.0)) / x_m) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(c_m * s_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2.8e-16)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (cos((x_m * -2.0)) / x_m) / (s_m * ((x_m * c_m) * (c_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2.8e-16], 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] / x$95$m), $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e-16

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 60.5%

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

      10. unpow2 60.5%

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

   3. Simplified 60.5%

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

      1. div-inv 60.5%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 60.4%

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

      4. pow-prod-down 60.4%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. *-commutative 42.4%

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

      9. pow-flip 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow-prod-down 52.3%

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

      12. sqrt-pow1 80.8%

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

      13. metadata-eval 80.8%

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

      14. pow1 80.8%

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

      15. *-commutative 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 71.9%

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

   8. Taylor expanded in x around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. *-commutative 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. swap-sqr 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*r* 85.9%

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

      12. unpow2 85.9%

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

      13. exp-to-pow 55.1%

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

      14. *-commutative 55.1%

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

      15. exp-neg 55.2%

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

      16. *-commutative 55.2%

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

      17. distribute-rgt-neg-in 55.2%

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

      18. metadata-eval 55.2%

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

      19. exp-to-pow 86.5%

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

   10. Simplified 85.3%

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

   if 2.8000000000000001e-16 < x

   1. Initial program 60.0%

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

      1. associate-/r* 59.6%

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

      2. cos-neg 59.6%

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

      3. distribute-rgt-neg-out 59.6%

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

      4. distribute-rgt-neg-out 59.6%

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

      5. *-commutative 59.6%

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

      6. distribute-rgt-neg-in 59.6%

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

      7. metadata-eval 59.6%

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

      8. *-commutative 59.6%

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

      9. associate-*l* 53.9%

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

      10. unpow2 53.9%

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

   3. Simplified 53.9%

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

      1. div-inv 53.9%

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

      2. add-sqr-sqrt 53.9%

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

      3. times-frac 53.9%

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

      4. pow-prod-down 53.9%

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

      5. sqrt-pow1 45.4%

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

      6. metadata-eval 45.4%

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

      7. pow1 45.4%

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

      8. *-commutative 45.4%

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

      9. pow-flip 45.4%

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

      10. metadata-eval 45.4%

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

      11. pow-prod-down 52.2%

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

      12. sqrt-pow1 78.0%

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

      13. metadata-eval 78.0%

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

      14. pow1 78.0%

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

      15. *-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. sqr-pow 78.0%

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

      2. associate-/l* 86.0%

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

      3. metadata-eval 86.0%

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

      4. unpow-1 86.0%

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

      5. metadata-eval 86.0%

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

      6. unpow-1 86.0%

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

   8. Applied egg-rr 86.0%

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

   10. Applied egg-rr 83.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{x}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-102}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot -2\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 3.5e-102)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (/ (cos (* x_m -2.0)) (* (* x_m c_m) (* x_m s_m))) (* c_m s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.5e-102) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (cos((x_m * -2.0)) / ((x_m * c_m) * (x_m * s_m))) / (c_m * s_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 3.5d-102) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = (cos((x_m * (-2.0d0))) / ((x_m * c_m) * (x_m * s_m))) / (c_m * s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 3.5e-102) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * -2.0)) / ((x_m * c_m) * (x_m * s_m))) / (c_m * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 3.5e-102:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = (math.cos((x_m * -2.0)) / ((x_m * c_m) * (x_m * s_m))) / (c_m * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 3.5e-102)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(x_m * c_m) * Float64(x_m * s_m))) / Float64(c_m * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 3.5e-102)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = (cos((x_m * -2.0)) / ((x_m * c_m) * (x_m * s_m))) / (c_m * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 3.5e-102], 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] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.49999999999999986e-102

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 59.9%

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

      10. unpow2 59.9%

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

   3. Simplified 59.9%

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

      1. div-inv 59.9%

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

      2. add-sqr-sqrt 59.8%

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

      3. times-frac 59.8%

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

      4. pow-prod-down 59.8%

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

      5. sqrt-pow1 44.3%

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

      6. metadata-eval 44.3%

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

      7. pow1 44.3%

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

      8. *-commutative 44.3%

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

      9. pow-flip 44.3%

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

      10. metadata-eval 44.3%

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

      11. pow-prod-down 55.1%

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

      12. sqrt-pow1 80.9%

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

      13. metadata-eval 80.9%

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

      14. pow1 80.9%

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

      15. *-commutative 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around 0 71.2%

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

   8. Taylor expanded in x around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow2 54.9%

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

      4. unpow2 54.9%

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

      5. swap-sqr 69.3%

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

      6. unpow2 69.3%

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

      7. swap-sqr 83.4%

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

      8. *-commutative 83.4%

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

      9. associate-*r* 82.9%

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

      10. *-commutative 82.9%

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

      11. associate-*r* 84.7%

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

      12. unpow2 84.7%

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

      13. exp-to-pow 54.3%

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

      14. *-commutative 54.3%

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

      15. exp-neg 54.4%

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

      16. *-commutative 54.4%

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

      17. distribute-rgt-neg-in 54.4%

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

      18. metadata-eval 54.4%

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

      19. exp-to-pow 85.3%

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

   10. Simplified 84.0%

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

   if 3.49999999999999986e-102 < x

   1. Initial program 61.2%

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

      1. associate-/r* 60.9%

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

      2. cos-neg 60.9%

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

      3. distribute-rgt-neg-out 60.9%

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

      4. distribute-rgt-neg-out 60.9%

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

      5. *-commutative 60.9%

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

      6. distribute-rgt-neg-in 60.9%

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

      7. metadata-eval 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*l* 56.1%

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

      10. unpow2 56.1%

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

   3. Simplified 56.1%

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

      1. div-inv 56.1%

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

      2. add-sqr-sqrt 56.1%

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

      3. times-frac 56.1%

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

      4. pow-prod-down 56.1%

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

      5. sqrt-pow1 41.3%

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

      6. metadata-eval 41.3%

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

      7. pow1 41.3%

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

      8. *-commutative 41.3%

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

      9. pow-flip 41.3%

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

      10. metadata-eval 41.3%

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

      11. pow-prod-down 46.9%

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

      12. sqrt-pow1 78.4%

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

      13. metadata-eval 78.4%

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

      14. pow1 78.4%

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

      15. *-commutative 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. sqr-pow 78.4%

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

      2. times-frac 83.8%

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

      3. metadata-eval 83.8%

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

      4. unpow-1 83.8%

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

      5. metadata-eval 83.8%

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

      6. unpow-1 83.8%

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

   8. Applied egg-rr 83.8%

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

      1. associate-*r* 84.8%

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

      2. associate-/l/ 84.7%

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

      3. un-div-inv 85.9%

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

      4. associate-/l/ 86.1%

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

      5. frac-times 86.1%

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

      6. *-commutative 86.1%

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

      7. *-un-lft-identity 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. *-commutative 86.1%

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

   10. Applied egg-rr 86.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-102}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot c\right) \cdot \left(x \cdot s\right)}}{c \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{\left(x\_m \cdot s\_m\right) \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 2.8e-16)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* (* x_m s_m) (* (* x_m c_m) (* c_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / ((x_m * s_m) * ((x_m * c_m) * (c_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.8d-16) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / ((x_m * s_m) * ((x_m * c_m) * (c_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 2.8e-16) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((x_m * s_m) * ((x_m * c_m) * (c_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 2.8e-16:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / ((x_m * s_m) * ((x_m * c_m) * (c_m * s_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 2.8e-16)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(x_m * s_m) * Float64(Float64(x_m * c_m) * Float64(c_m * s_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 2.8e-16)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / ((x_m * s_m) * ((x_m * c_m) * (c_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 2.8e-16], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e-16

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

      2. cos-neg 67.0%

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

      3. distribute-rgt-neg-out 67.0%

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

      4. distribute-rgt-neg-out 67.0%

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

      5. *-commutative 67.0%

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

      6. distribute-rgt-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*l* 60.5%

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

      10. unpow2 60.5%

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

   3. Simplified 60.5%

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

      1. div-inv 60.5%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 60.4%

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

      4. pow-prod-down 60.4%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. *-commutative 42.4%

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

      9. pow-flip 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow-prod-down 52.3%

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

      12. sqrt-pow1 80.8%

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

      13. metadata-eval 80.8%

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

      14. pow1 80.8%

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

      15. *-commutative 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 71.9%

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

   8. Taylor expanded in x around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. *-commutative 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. swap-sqr 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*r* 85.9%

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

      12. unpow2 85.9%

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

      13. exp-to-pow 55.1%

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

      14. *-commutative 55.1%

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

      15. exp-neg 55.2%

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

      16. *-commutative 55.2%

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

      17. distribute-rgt-neg-in 55.2%

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

      18. metadata-eval 55.2%

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

      19. exp-to-pow 86.5%

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

   10. Simplified 85.3%

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

   if 2.8000000000000001e-16 < x

   1. Initial program 60.0%

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

      1. associate-/r* 59.6%

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

      2. cos-neg 59.6%

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

      3. distribute-rgt-neg-out 59.6%

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

      4. distribute-rgt-neg-out 59.6%

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

      5. *-commutative 59.6%

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

      6. distribute-rgt-neg-in 59.6%

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

      7. metadata-eval 59.6%

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

      8. *-commutative 59.6%

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

      9. associate-*l* 53.9%

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

      10. unpow2 53.9%

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

   3. Simplified 53.9%

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

      1. div-inv 53.9%

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

      2. add-sqr-sqrt 53.9%

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

      3. times-frac 53.9%

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

      4. pow-prod-down 53.9%

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

      5. sqrt-pow1 45.4%

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

      6. metadata-eval 45.4%

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

      7. pow1 45.4%

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

      8. *-commutative 45.4%

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

      9. pow-flip 45.4%

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

      10. metadata-eval 45.4%

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

      11. pow-prod-down 52.2%

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

      12. sqrt-pow1 78.0%

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

      13. metadata-eval 78.0%

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

      14. pow1 78.0%

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

      15. *-commutative 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. sqr-pow 78.0%

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

      2. associate-/l* 86.0%

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

      3. metadata-eval 86.0%

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

      4. unpow-1 86.0%

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

      5. metadata-eval 86.0%

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

      6. unpow-1 86.0%

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

   8. Applied egg-rr 86.0%

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

   10. Applied egg-rr 80.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.4% 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 65.0%

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

   1. associate-/r* 64.9%

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

   2. cos-neg 64.9%

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

   3. distribute-rgt-neg-out 64.9%

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

   4. distribute-rgt-neg-out 64.9%

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

   5. *-commutative 64.9%

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

   6. distribute-rgt-neg-in 64.9%

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

   7. metadata-eval 64.9%

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

   8. *-commutative 64.9%

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

   9. associate-*l* 58.6%

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

   10. unpow2 58.6%

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

3. Simplified 58.6%

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

   1. div-inv 58.6%

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

   2. add-sqr-sqrt 58.5%

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

   3. times-frac 58.5%

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

   4. pow-prod-down 58.5%

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

   5. sqrt-pow1 43.2%

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

   6. metadata-eval 43.2%

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

   7. pow1 43.2%

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

   8. *-commutative 43.2%

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

   9. pow-flip 43.3%

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

   10. metadata-eval 43.3%

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

   11. pow-prod-down 52.3%

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

   12. sqrt-pow1 80.0%

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

   13. metadata-eval 80.0%

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

   14. pow1 80.0%

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

   15. *-commutative 80.0%

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

6. Applied egg-rr 80.0%

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

7. Taylor expanded in x around 0 65.9%

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

8. Taylor expanded in x around 0 53.2%

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

   1. *-commutative 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 63.6%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)} \cdot {c}^{2}} \]

   6. unpow2 63.6%

      \[\leadsto \frac{1}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]

   7. swap-sqr 75.6%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot c\right)}} \]

   8. *-commutative 75.6%

      \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot c\right)} \]

   9. associate-*r* 75.2%

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right)} \cdot \left(\left(x \cdot s\right) \cdot c\right)} \]

   10. *-commutative 75.2%

       \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot c\right)} \]

   11. associate-*r* 76.7%

       \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

   12. unpow2 76.7%

       \[\leadsto \frac{1}{\color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}} \]

   13. exp-to-pow 47.5%

       \[\leadsto \frac{1}{\color{blue}{e^{\log \left(s \cdot \left(x \cdot c\right)\right) \cdot 2}}} \]

   14. *-commutative 47.5%

       \[\leadsto \frac{1}{e^{\color{blue}{2 \cdot \log \left(s \cdot \left(x \cdot c\right)\right)}}} \]

   15. exp-neg 47.6%

       \[\leadsto \color{blue}{e^{-2 \cdot \log \left(s \cdot \left(x \cdot c\right)\right)}} \]

   16. *-commutative 47.6%

       \[\leadsto e^{-\color{blue}{\log \left(s \cdot \left(x \cdot c\right)\right) \cdot 2}} \]

   17. distribute-rgt-neg-in 47.6%

       \[\leadsto e^{\color{blue}{\log \left(s \cdot \left(x \cdot c\right)\right) \cdot \left(-2\right)}} \]

   18. metadata-eval 47.6%

       \[\leadsto e^{\log \left(s \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{-2}} \]

   19. exp-to-pow 77.1%

       \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}} \]

10. Simplified 76.0%

    \[\leadsto \color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]

11. Final simplification 76.0%

    \[\leadsto {\left(c \cdot \left(x \cdot s\right)\right)}^{-2} \]
12. Add Preprocessing

Alternative 10: 78.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m)))) (/ (/ 1.0 t_0) t_0)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	return (1.0 / t_0) / t_0;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (x_m * c_m)
    code = (1.0d0 / t_0) / t_0
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	return (1.0 / t_0) / t_0;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	return (1.0 / t_0) / t_0

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 64.9%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. cos-neg 64.9%

      \[\leadsto \frac{\frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   3. distribute-rgt-neg-out 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot \left(-x\right)\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   4. distribute-rgt-neg-out 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(-2 \cdot x\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   5. *-commutative 64.9%

      \[\leadsto \frac{\frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   6. distribute-rgt-neg-in 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   7. metadata-eval 64.9%

      \[\leadsto \frac{\frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   8. *-commutative 64.9%

      \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x} \]

   9. associate-*l* 58.6%

      \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot \left(x \cdot x\right)}} \]

   10. unpow2 58.6%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot \color{blue}{{x}^{2}}} \]

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 58.6%

      \[\leadsto \frac{\color{blue}{\cos \left(x \cdot -2\right) \cdot \frac{1}{{c}^{2}}}}{{s}^{2} \cdot {x}^{2}} \]

   2. add-sqr-sqrt 58.5%

      \[\leadsto \frac{\cos \left(x \cdot -2\right) \cdot \frac{1}{{c}^{2}}}{\color{blue}{\sqrt{{s}^{2} \cdot {x}^{2}} \cdot \sqrt{{s}^{2} \cdot {x}^{2}}}} \]

   3. times-frac 58.5%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{\sqrt{{s}^{2} \cdot {x}^{2}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}}} \]

   4. pow-prod-down 58.5%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\sqrt{\color{blue}{{\left(s \cdot x\right)}^{2}}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   5. sqrt-pow1 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   6. metadata-eval 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\left(s \cdot x\right)}^{\color{blue}{1}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   7. pow1 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{s \cdot x}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   8. *-commutative 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{x \cdot s}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   9. pow-flip 43.3%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{\color{blue}{{c}^{\left(-2\right)}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   10. metadata-eval 43.3%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{\color{blue}{-2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   11. pow-prod-down 52.3%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\sqrt{\color{blue}{{\left(s \cdot x\right)}^{2}}}} \]

   12. sqrt-pow1 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{2}{2}\right)}}} \]

   13. metadata-eval 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{{\left(s \cdot x\right)}^{\color{blue}{1}}} \]

   14. pow1 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{s \cdot x}} \]

   15. *-commutative 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{x \cdot s}} \]

6. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{x \cdot s}} \]

7. Taylor expanded in x around 0 65.9%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot s} \cdot \frac{{c}^{-2}}{x \cdot s} \]
8. Step-by-step derivation

   1. metadata-eval 65.9%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{{c}^{\color{blue}{\left(-1 + -1\right)}}}{x \cdot s} \]

   2. pow-prod-up 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{{c}^{-1} \cdot {c}^{-1}}}{x \cdot s} \]

   3. inv-pow 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{\frac{1}{c}} \cdot {c}^{-1}}{x \cdot s} \]

   4. inv-pow 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\frac{1}{c} \cdot \color{blue}{\frac{1}{c}}}{x \cdot s} \]

   5. associate-*r/ 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \color{blue}{\left(\frac{1}{c} \cdot \frac{\frac{1}{c}}{x \cdot s}\right)} \]

   6. associate-*l/ 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \color{blue}{\frac{1 \cdot \frac{\frac{1}{c}}{x \cdot s}}{c}} \]

   7. *-un-lft-identity 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{\frac{\frac{1}{c}}{x \cdot s}}}{c} \]

   8. frac-times 76.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{1}{c}}{x \cdot s}}{\left(x \cdot s\right) \cdot c}} \]

   9. *-un-lft-identity 76.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{c}}{x \cdot s}}}{\left(x \cdot s\right) \cdot c} \]

   10. associate-/l/ 76.0%

       \[\leadsto \frac{\color{blue}{\frac{1}{\left(x \cdot s\right) \cdot c}}}{\left(x \cdot s\right) \cdot c} \]

   11. *-commutative 76.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(s \cdot x\right)} \cdot c}}{\left(x \cdot s\right) \cdot c} \]

   12. associate-*r* 75.6%

       \[\leadsto \frac{\frac{1}{\color{blue}{s \cdot \left(x \cdot c\right)}}}{\left(x \cdot s\right) \cdot c} \]

   13. *-commutative 75.6%

       \[\leadsto \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{\color{blue}{\left(s \cdot x\right)} \cdot c} \]

   14. associate-*r* 77.1%

       \[\leadsto \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{\color{blue}{s \cdot \left(x \cdot c\right)}} \]

9. Applied egg-rr 77.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}} \]
10. Add Preprocessing

Alternative 11: 77.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{1}{c\_m}}{\left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right) \cdot \left(x\_m \cdot s\_m\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ 1.0 c_m) (* (* s_m (* x_m c_m)) (* x_m s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (1.0 / c_m) / ((s_m * (x_m * c_m)) * (x_m * s_m));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / c_m) / ((s_m * (x_m * c_m)) * (x_m * s_m))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return (1.0 / c_m) / ((s_m * (x_m * c_m)) * (x_m * s_m));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (1.0 / c_m) / ((s_m * (x_m * c_m)) * (x_m * s_m))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(1.0 / c_m) / Float64(Float64(s_m * Float64(x_m * c_m)) * Float64(x_m * s_m)))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = (1.0 / c_m) / ((s_m * (x_m * c_m)) * (x_m * s_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 64.9%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]

   2. cos-neg 64.9%

      \[\leadsto \frac{\frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   3. distribute-rgt-neg-out 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot \left(-x\right)\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   4. distribute-rgt-neg-out 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(-2 \cdot x\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   5. *-commutative 64.9%

      \[\leadsto \frac{\frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   6. distribute-rgt-neg-in 64.9%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   7. metadata-eval 64.9%

      \[\leadsto \frac{\frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x} \]

   8. *-commutative 64.9%

      \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x} \]

   9. associate-*l* 58.6%

      \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot \left(x \cdot x\right)}} \]

   10. unpow2 58.6%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot \color{blue}{{x}^{2}}} \]

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 58.6%

      \[\leadsto \frac{\color{blue}{\cos \left(x \cdot -2\right) \cdot \frac{1}{{c}^{2}}}}{{s}^{2} \cdot {x}^{2}} \]

   2. add-sqr-sqrt 58.5%

      \[\leadsto \frac{\cos \left(x \cdot -2\right) \cdot \frac{1}{{c}^{2}}}{\color{blue}{\sqrt{{s}^{2} \cdot {x}^{2}} \cdot \sqrt{{s}^{2} \cdot {x}^{2}}}} \]

   3. times-frac 58.5%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{\sqrt{{s}^{2} \cdot {x}^{2}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}}} \]

   4. pow-prod-down 58.5%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\sqrt{\color{blue}{{\left(s \cdot x\right)}^{2}}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   5. sqrt-pow1 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   6. metadata-eval 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{{\left(s \cdot x\right)}^{\color{blue}{1}}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   7. pow1 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{s \cdot x}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   8. *-commutative 43.2%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{\color{blue}{x \cdot s}} \cdot \frac{\frac{1}{{c}^{2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   9. pow-flip 43.3%

      \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{\color{blue}{{c}^{\left(-2\right)}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   10. metadata-eval 43.3%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{\color{blue}{-2}}}{\sqrt{{s}^{2} \cdot {x}^{2}}} \]

   11. pow-prod-down 52.3%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\sqrt{\color{blue}{{\left(s \cdot x\right)}^{2}}}} \]

   12. sqrt-pow1 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{2}{2}\right)}}} \]

   13. metadata-eval 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{{\left(s \cdot x\right)}^{\color{blue}{1}}} \]

   14. pow1 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{s \cdot x}} \]

   15. *-commutative 80.0%

       \[\leadsto \frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{\color{blue}{x \cdot s}} \]

6. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{x \cdot s} \cdot \frac{{c}^{-2}}{x \cdot s}} \]

7. Taylor expanded in x around 0 65.9%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot s} \cdot \frac{{c}^{-2}}{x \cdot s} \]
8. Step-by-step derivation

   1. metadata-eval 65.9%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{{c}^{\color{blue}{\left(-1 + -1\right)}}}{x \cdot s} \]

   2. pow-prod-up 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{{c}^{-1} \cdot {c}^{-1}}}{x \cdot s} \]

   3. inv-pow 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{\frac{1}{c}} \cdot {c}^{-1}}{x \cdot s} \]

   4. inv-pow 65.8%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\frac{1}{c} \cdot \color{blue}{\frac{1}{c}}}{x \cdot s} \]

   5. associate-*r/ 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \color{blue}{\left(\frac{1}{c} \cdot \frac{\frac{1}{c}}{x \cdot s}\right)} \]

   6. frac-times 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \color{blue}{\frac{1 \cdot \frac{1}{c}}{c \cdot \left(x \cdot s\right)}} \]

   7. *-un-lft-identity 74.0%

      \[\leadsto \frac{1}{x \cdot s} \cdot \frac{\color{blue}{\frac{1}{c}}}{c \cdot \left(x \cdot s\right)} \]

   8. frac-times 72.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   9. *-un-lft-identity 72.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{c}}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]

   10. associate-*r* 72.5%

       \[\leadsto \frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}} \]

   11. *-commutative 72.5%

       \[\leadsto \frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)} \]

   12. *-commutative 72.5%

       \[\leadsto \frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

9. Applied egg-rr 72.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

10. Final simplification 72.5%

    \[\leadsto \frac{\frac{1}{c}}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot s\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))